College Algebra

Chapter 2: Geometry

Geometry is the mathematical study of the relationships between collections of points, curves, angles, surfaces and solid objects including the measurement of these objects and the distances between them. This review will discuss the correspondence between the real numbers and the points of a line, the general idea of analytic geometry, as well as the equations of lines, circles, and parabolas as well as the measurement of distance between points.

2.1 The Pythagorean Theorem

The most important result of classical synthetic geometry is:

Theorem 1.1: (Pythagorean Theorem) If is a right triangle with hypotenuse of length c and legs of length a and b, then .

To understand its proof, we need to know

Proposition 1.1: The sum of the angles of any triangle is 180 degrees.

Proof: Let be an arbitrary triangle. Construct a line DE parallel to the base BC and through the third vertex A as shown in the diagram below.

Since BA is transversal to the pair of parallel lines BC and DE, one has . Similarly, CA is transversal to the pair of parallel lines BC and DE. So, . Since DAE is a straight line, we have . From the diagram, we see that

as was to be shown.

The proof of the Pythagorean Theorem will also require some simple facts about the areas of squares and triangles. For future reference, let's state these and other results as

Proposition 1.2:(Areas and Perimeters)

1. The area of a rectangle (or even a parallelogram) is the product of the lengths of its base and its height.
2. The area of a triangle is half the product of the lengths of its base and its height.
3. The area of a circle is where r is its radius. The circumference (perimeter) of of a circle is .

Now it is easy to see why the Pythagorean Theorem is true. Start with an arbitrary right triangle where C is the right angle and the opposite side (the hypotenuse) is of length c. Let a and b be the lengths of the legs which are opposite to the angles A and B respectively.

1. Construct the figure below where the square in the middle has sides of length c. On each of its sides, place a triangle congruent to using the side of the square as the hypotenuse.
2. The interior angles of the square are, of course, right angles and the sum of the angles opposite the legs of each of the triangles is by Proposition 1.1. It follows that the legs of the triangles at each of the vertices of the square actually form a straight line segment (because the sum of the three angles there is precisely 180 degrees. So our figure is actually a large square with sides of length a + b.
3. Now the large square can also be considered to be the union of the smaller square with side c and four congruent right triangles with base a and height b. So we can calculate the area of the figure in two ways (using Proposition 1.2) and equate the answers: . When you multiply out and simplify this equation, one finds , as was to be proved.

There are at least hundreds of extant proofs of the Pythagorean Theorem. The proof given here is quite different from the proof appearing in Euclid's Elements.

Corollary 1.1: i. The length of the diagonal of a rectangle with sides of lengths a and b is precisely .

ii. Let T be a triangle with sides of length a, b, and c opposite angles A, B, and C respectively. Then T is a right triangle with right angle C if .

Proof: i. The first assertion is an obvious consequence of Theorem 1.

ii. Let T' be a triangle where angle C' is a right angle and the sides opposite angles A' and B' are of length a and b respectively. By Theorem 1, the length of the side opposite angle C' is of length . Then corresponding sides of the triangles T and T' are of equal length; so the two triangles are congruent. But then angle C must be equal to angle C' and so angle C is a right angle.

2.2 Trigonometry

We will need to know about similar triangles. Two triangles and are defined to be similar if , , and . By Proposition 1.1, if two of these equations hold, then so does the third. The result we need from geometry is:

Proposition 2.1: If the two triangles and are similar, then the ratio of the lengths of a pair of sides of one triangle is equal to the ratio of the lengths of the corresponding pair of sides of the other triangle, e.g. . Conversely, if the ratios of the lengths of sides of one triangle are equal to the ratios of the lengths of the corresponding sides of another triangle, then the triangles are similar.

Angle Measurement: Angles are measured in degrees or in radians. Start with the unit circle centered at the origin (0, 0). Make one side of the angle the positive x-axis. The other side of the angle is another ray from the origin as shown in the diagram below. We can measure the angle as a portion of the circle swept out by the angle with the whole circle being 360 degrees. In the diagram, the angle t sweeps out about a sixth of the circle so, it is about 360/6 = 60 degrees. Alternatively, we can measure the angle as the length of the arc swept out by the angle. The entire circumference of the unit circle is units. In the diagram, the angle t sweeps out the red arc which is about 1/6 the entire circumference or radians.

Radians and degrees are just different units for measuring angles. To convert between them, just remember that 360 degrees is radians (because both represent the whole circle). The formula for conversion is then:

If you use a circle of radius , then the arc swept out by the angle is times as long as the arc swept out on a circle of radius 1. So, if the angle is radians, the arc length L is is rt units:

In the diagram, the angle was swept out in a counterclockwise direction. If one sweeps it out in a clockwise direction, then the measure of the angle is a negative number. Its magnitude is the same as that of the same arc traversed counterclockwise; but the sign is made negative.

Referring to the diagram above, the cosine and sine of the angle t are defined to be the the coordinates of the point on the unit circle where the radius line meets the circle. If A is the point on the unit circle, then . Since A lies on the unit circle, the the radius is 1 and so the Pythagorean Theorem tells that

. Further, it is clear by the symmetry of the circle that we have:

These properties are often described by saying that sine is an odd function and that cosine is an even function.

Also, by symmetry, one has

The angle is called the complement of the angle and the two angles are said to be complementary. The first identity says that the cosine is the sine of the complement; this was the original historical meaning of cosine -- it was an abbreviated form of complementary sine.

Here are the graphs of the sine and cosine functions. You should be able to see the symmetry properties of the functions in the graphs.

If B is the point on the ray 0A where the ray intersects the circle of radius r: , then is equal to the ratio of the x-coordinate of B and the the hypotenuse r (because of similar triangles). This explains the common definition of the cosine as being the ratio of the adjacent side to the hypotenuse; when using this, be careful, that that the adjacent side may be either a positive or negative number. Similarly, the sine of the same angle is the quotient of the opposite side over the hypotenuse (where the opposite side might be negative). These definitions allow one to create tables of trigonometric functions for practical use in solving right triangles. For example, Ptolemy computed tables in half degree increments.

The remaining trigonometric functions are:

Geometrically, the tangent function is the y-coordinate of the point A in the diagram:

The graphs of these functions are:

Example 2.1: Dividing the identity , by and using the above definitions, one obtains the identity:

Similarly, by dividing by , one obtains the identity:

Example 2.2: Let be the angle opposite the side of length 3 in a right triangle whose sides are of length 3, 4, and 5. Then the trigonometric functions of are:

The values of the trigonometric functions at a number of commonly occurring angles are given in the table below:

Degrees	Radians	sine	cosine	tangent	cotangent	secant	cosecant
0	0	0	1	0	undefined	1	undefined
30		1/2					2
45				1	1
60			1/2			2
90		1	0	undefined	0	undefined	1

Rather than memorize the table, it is easier to keep in mind the figure above used to define the trig functions and reconstruct the two standard triangles shown below. In the left triangle, one has a 30-60-90 degree triangle with the point D at the midpoint of the hypotenuse and being equilateral. So, if the height is 1, the hypotenuse is 2 and the base has length by the Pythagorean Theorem. The triangle on the right is 45-45-90 degrees. The two legs are of equal length; if they have length 1, the hypotenuse has length by the Pythagorean Theorem.

The Pythagorean Theorem only works for right triangles. The next result allows one to solve for the sides and angles of arbitrary triangles:

Proposition 2.2: Let be any triangle and a, b, c be the lengths of the sides opposite angles A, B, and C respectively. Then

1. ( Law of Cosines) .
2. (Law of Sines) .

Proof: (i) The law of cosines is a generalization of the Pythagorean Theorem to arbitrary triangles. To prove it, we need to create some right triangles; drop a line from A perpendicular to BC and let D be the point where the perpendicular intersects the line BC. There are three cases depending on whether D is between B and C or on one of the two infinite rays outside of the BC. We'll consider one case here and leave it for the reader to construct proofs of the remaining cases. Suppose that D is between B and C. Let h, r, and s be the lengths of AD, CD, and DB respectively. Then a = r + s, cos(C) = r/b and . The right side of the law of cosines is therefore:

The other two cases are similar using a = r - s or a = s - r in place of a = r + s.

(ii) Use the same construction as in part (i) and treat three cases. In case D is between B and C, one has sin(C) = h/b and sin(B) = h/c. So, sin(C)/c = (h/b)/c = (h/c)/b = sin(B)/b as desired.

The following result will be the basis for a geometrical description of the multiplication of complex numbers:

Proposition 2.3:(Addition Formulas) For arbitrary angles and , one has:

1. .
2. .
3. .
4. .

Proof: Using the fact that sine is odd and cosine is even, it is clear that the first two equations and the last two equations are equivalent. Furthermore, the identity allows one to replace assertions about cosine with ones about sine; so, one can check that any one of the four assertions easily gives the remaining three.

Let's try to show the last equation. We measure the angles s, t and s-t from the positive x-axis. Let A, B, and C be the points on the unit circle where the second ray of the angles s, t, and s - t respectively intersect the unit circle. Let D = (1, 0) be the point where the positive x-axis intersects the unit circle. Then the arcs AB and CD are equal in length and so are lengths of the chords AB and CD. One has A = (cos(s), sin(s)), B = (cos(t), sin(t)), and C = (cos(s-t), sin(s-t)). Using the distance formula, one sees that AB = CD means that

If you multiply this out and use the Pythagorean Theorem to simplify, the result is the fourth equation.

2.3 Polar Coordinates and Transformations

Cartesian coordinates are one way of giving names to points in the plane. The diagram

The relationships between the polar coordinates (r, t) and the cartesian coordinates (x, y) of a given point P are:

Example 3.1: If the cartesian coordinates of P are (3, 2), then the polar coordinates of P are (r, t) where and . A calculator can be used to see that the angle whose tangent is 2/3 is approximately 0.6 radians or 34 degrees. Since (3, 2) is in the first quadrant, we see that t = 34 degrees. In general, you need to be careful to modify the value obtained by your calculator. The tangent function repeats itself every 180 degrees; your calculator always gives a value between -90 degrees and 90 degrees; so, depending on the quadrant of the point, you may or may not need to add 180 to the value given by your calculator.

If a point Q has polar coordinates (5,1), then its cartesian coordinates are x = r cos(t) = 5 cos(1) and y = r sin(t) = 5 sin(1). A calculator shows that this is approximately (2.7, 4.2). This is much easier; but you still need to check the units of the angle (degrees or radians) and make sure that your calculator is set to evaluate the trig functions with angles measured in this unit.

Geometric Interpretation of operations on complex numbers. Recall that every complex number can be written as z = x + i y where . The real numbers x and y are called the real and imaginary parts of the complex number z. By associating z with the point (x, y) in Cartesian coordinates, we can interpret complex numbers as being simply points in the plane. Our goal here is to check to interpret geometrically the arithmetic operations on complex numbers.

Given two complex numbers z = x + i y and w = u + i v, their sum is simply z + w = (x + u) + i (y + v). If you sketch the corresponding points, you see that the points (0,0), z, w, and z + w are the vertices of a parallelogram. So addition of complex numbers corresponds to the parallelogram rule for adding vectors. This is exactly the way you added forces in your physics course.

Given two complex number z = x + i y and w = u + i v, the product zw has to be the same as what we would get by using the distributive law together with . So, the product is

To understand this, let's write z = (r,t), w = (p,q) where the coordinates are now polar. One has x = r cos(t), y = r sin(t), u = p cos(q), and v = p sin(t). Substituting into our formula for the product zw gives:

where the last simplification used the addition formulas for sine and cosine. We have seen that the product zw has polar coordinates (rp, t + q).

If z = x + iy is a complex number with polar coordinates (r, t), then r is called the length or absolute value of z and the angle t is called the argument of z.

Proposition 3.1: The product of two complex numbers has length equal to the product of the lengths of the factors and argument the sum of arguments of the factors. To divide two complex numbers, divide the lengths and subtract the arguments.

Rotations of Graphs In particular, if we wanted to rotate a complex number w = u + iv through an angle t, then we would just multiply it by z = cos(t) = i sin(t). This would keep the length the same because |z| = 1 but increase the argument by t. The same would work for arbitrary graphs, e.g. suppose we have the graph of all points which satisfy f(x, y) = 0. Then rotate x + iy through an angle -t by multiplying by cos(-t) + i sin(-t). This gives the complex number (x cos(t) + y sin(t) + i (y cos(t) - x sin(t)). Substitute these into the original equation to get

If P = (x,y) satisfies this last equation, then the point obtained by rotating P -t radians satisfies the original equation f(x, y) = 0. So, we see have seen that our new equation is the equation of the original graph rotated t radians.

Corollary 3.1: Given the equation f(x,y) = 0 for the graph of a curve, the equation for the curve obtained by rotating the original curve through an angle of t radians is obtained by replacing the variables x + i y with the new variables u + i v = (x + i y)(cos(-t) + i sin(-t)).

The preceding was the most difficult of the common transformations of the plane. For convenience, we collect together here the analogous rules for handling translations (shifts), expansions (dilations), and reflections.

Translation of Graphs If the point (x,y) satisfies the equation f(x, y) = 0, and if A and B are real numbers, then the point (x + A, y + B) satisfies the equation f(x - A, y - B) = 0 (because f((x+A) - A, (y+B) - B) = f(x, y) = 0). Another way of saying this is: if the graph of a function y = f(x) is moved A units to the right and B units to the right, then one obtains the graph of y - B = f(x - A). This amounts to the simple rules:

Graph Compression and Expansion If the point (x,y) satisfies the equation f(x, y) = 0, and if A and B are non-zero real numbers, then the point (Ax, By) satisifes the equation f(x/A, y/B) = 0 (because f((Ax)/A, (By)/B) = f(x, y) = 0). Another way of saying this is: if the graph of a function y = f(x) is expanded by a factor of A horizontally and by a factor of B vertically, then the result is the graph of y = Bf(x/A). Again, you can give simple rules:

Reflecting a graph across an axis If the point (x, y) satisfies the equation f(x, y) = 0, then the point (-x, y) obtained by reflecting the point (x, y) across the y-axis satisfies f(-x, y) = 0. Similarly, the point (x, -y) obtained by reflecting the point (x, y) across the x-axis satsifes the equation f(x, -y) = 0.

Existence of Roots of complex numbers Let z be an arbitrary non-zero complex number and n be a positive integer. Then z has at least n complex roots. In fact, if z has length |z| and argument t, then Proposition 3.1 says that the complex number with length and argument t/n must be an root of z. Furthermore, let . Then |w| = 1 and w has argument . So, again by Proposition 3.1, ; in fact, every power of w also has power equal to 1. It follows that each of the numbers for k = 0, 1, ..., n - 1 are roots of z. We know that they are all distinct because they all have different arguments; so our assertion is proved. In a later chapter, we will see that there are no other roots of z.

2.4 Lines, Circles, and Parabolas

Lines, circles, and other curves like parabolas should simply be the sets of solutions of certain algebraic equations. This appears to be the case. For example, we can define a vertical line to be the set of solutions of some equation x = a, where a is a constant. Similarly, a horizontal line is the set of solutions of an equation of the form for some constant b.

Definition 4.1: A line is either a horizontal or vertical line or else it is the set of solutions of an equation of the form y = mx + b where m and b are constants.

The equation y = mx + b (or x = a) is called the equation of the line which is the set of solutions of the equation. Note that this is a 1-1 correspondence between the set of lines and the set of equations of lines. In particular, different equations correspond to different lines; in fact, you can easily convince yourself that through any two points there is precisely one line. The constant m is called the slope of the line and the constant b is called the y-intercept of the line. Note that the y-intercept is the y-coordinate of the point where the line intersects the y-axis. The slope and y-intercept of a vertical line is not defined. There are many algebraically equivalent forms of the equation of a line; the particular one y = mx + b is called the slope-intercept form of the equation of the line.

If for i = 1, 2 are two distinct points on the line y = mx + b, then one has for i = 1, 2 and, taking differences, one gets

Solving for m gives:

Proposition 4.1: The slope of the non-vertical line through and is given by .

In terms of trigonometry, the slope is the tangent of the angle the line makes with the x-axis. In fact, one can consider the lines to be the curves one gets by rotating and translating the x-axis. The equation of the x-axis is y = 0. If one rotates this through an angle of t radians, then the new equation is -x sin(t) + y cos(t) = 0 by Corollary 2 of the last section. This is either y = x tan(t) if cos(t) is non-zero or x = 0 (corresponding to a vertical line through the origin). Shifting vertically by b and horizontally by a gives y = (x - a) tan(t) + b or x = a as the general equation. In particular, the slope of the line is m = tan(t) (or undefined in case of a vertical line).

Example 4.2: The equation of the line through the two points and is given by

This is the so-called 2 point form of the equation of the line. It is proved by substituting in the two points.

Example 4.3: The equation of the line with slope m which passes through the point is . This is called the point-slope form of the equation of the line.

Example 4.4: Consider the line through the two points (2, 3) and (5,7). Its slope is . Its equation in point-slope form is (or equivalently ). To find the y-intercept, just substitute x = 0 into either of these equations and solve for y; the y-intercept is 1/3. So the slope-intercept form of the equation of the line is y = (4/3)x + 1/3. To find points on this line, just substitute in arbitrarily chosen values of x and solve for the y-coordinate.

Two lines which have no point of intersection are said to be parallel.

Example 4.5: Two distinct lines are parallel if and only if they are either both vertical or else they both have the same slope.

Suppose the two lines are not vertical. Let and be their equations and let's assume (x, y) is a point of intersection. Then it satisfies both equations. Equating them, we get and so . If the slopes were unequal, the coefficient of x would be non-zero and we could solve for and substitute this back into either of the original equations to obtain the value for y. If the slopes are not equal, it is easy to then verify that this pair is indeed a point of intersection, i.e. satisfies both equations. On the other hand, if the slopes were equal, then we would have which would mean that the lines were not distinct (because they have the same equation).

In case the two lines are vertical, the equations are of the form and where . Clearly, no pair (x, y) could satisfy both of these equations; so distinct vertical lines are parallel.

Finally, if one line has the equation and the other the equation , then the point (a, ma + b) is a point of intersection. This completes the proof of the assertion.

We say that two lines are perpendicular if they intersect at right angles to each other.

Example 4.6: Two lines are penpendicular if and only if one of the following is true:

1. One line has slope 0 and the other is vertical.
2. Neither line is vertical and the product of the slopes of the lines is -1.

If the two lines are perpendicular and neither is vertical, then one makes an angle of t radians with the x-axis and the other makes an angle of radians with the x-axis. Using the addition formulas for sine and cosine, one gets:

So, the second condition is true. If one of the is vertical, then the other is horizontal and the first condition is true. For the converse, one can verify that using the addition formulas that tan(t)tan(t+u) = -1 leads to a contradiction unless u is a multiple of radians.

A circle with center (a, b) and radius r is the set of points (x,y) whose distance from (a, b) is precisely r. By the distance formula, this circle is the set of solutions of

The set of solutions of an equation of the form where , b, and c are constants is called a parabola. The following properties are easy to verify:

1. If a > 0, the parabola opens upward; if a < 0, it opens downward.
2. If a > 0, the lowest point on the parabola (i.e. the one with smallest y-coordinate) is at (b, c). This point is called the vertex of the parabola.
3. The parabola is symmetric about its axis x = b (i.e. For every real number z, the points on the parabola with x-coordinates have the same y-coordinates).

Example 4.7: Find the equation of the line which passes through the two points of intersection of the circle centered at the origin of radius 1 and the circle centered at (1,1) with radius 1.

The equations of the two circles are and . If (x, y) is a point of intersection, it must satisfy both equations. Expanding out the second equation, one gets . Subtracting the first equation and simplifying gives the equation x + y = 1. If we knew that there were two points of intersection, then they would both satisfy this equation and it is the equation of a line; so it must be the desired line. (In order to verify that there are indeed two points of intersection, finish solving the system of equations and verify that the two solutions satisfy the original equations.)

2.5 Conic Sections

Let L be a non-vertical line through the origin. By rotating the line L around the y-axis, one obtains a pair of infinite cones with vertex at the origin. By intersecting this pair of cones with various planes, one obtains curves called ellipses, parabolas, and hyperbolas as well as some degenerate cases such as a single point or a single line. These intersections were called sections and so ellipses, parabolas, and hyperbolas were referred to as conic sections, i.e. sections of a cone.

Nowadays the importance of conic sections is not that they arise by intersecting a cone with a plane, but rather that they can be used to categorize all curves whose equations are polynomials of degree 2. So, lines are the curves represented by equations which are polynomials of degree 1, and conic sections are the curves represented by equations which are polynomials of degree 2.

In this section, we will define ellipses, parabolas, and hyperbolas without reference to sections of cones, and obtain equations for each in some special cases.

2.5.1 Parabolas

We have already defined parabolas to be the graphs of functions of the form . Let us give a more traditional definition:

Definition 5.1: Let F be a point and D be a line in the plane which does not contain F. Then the parabola with focus F and directrix D is the set of points P = (x, y) such that the distance from P to F is the same as the distance from P to D. (By the distance from P to the line D, we mean the shortest distance from P to any point of D, i.e. the length of the line segment obtained by dropping a line through P perpendicular to D.)

Now, let's look at a special case in which the vertex F = (0, d) and the line D is y = -d. If P = (x,y) is on the parabola with focus F and directrix D, then the distance formula tells us that

When you multiply this out and collect terms, one sees that this is just or .

Given the parabola, , we see that a = 1/(4d) and so the focus is at (0, d) = (0, 1/(4a)) and the directrix is y = -d = -1/(4a). Where is the focus and directrix of ?

2.5.2 Ellipses

One could define ellipses and hyperbolas in a manner exactly analogous to Definition 5.1, viz.

Definition 5.1a: Let F be a point and D be a line in the plane which does not contain F and e be a positive real number. Then the conic section with focus F, directrix D, and eccentricity e is the set of points P = (x, y) such that the distance from P to F is equal to e times the distance from P to D. If e = 1, the conic section is called a parabola. The conic section is called an ellipse if e < 1 and a hyperbola if e > 1.

Such a definition leads to rather complicated formulas, and so, instead, we will define

Definition 5.2: Let and be two not necessarily distinct points in the plane and let a be a positive real number. Then the ellipse with foci and and semimajor axis a is the set of points P = (x, y) such that the sum of the distances from P to the foci is exactly 2a.

Clearly, if a is less than half the distance between the foci, then the ellipse with semimajor axis a is the empty set.

Again, let us consider a special case: Let and where is a real number. Let a be a real number greater than c. Then if P = (x, y) is a point of the ellipse with these points as foci and with semimajor axis a, then we have by the distance formula:

This is the equation of the ellipse, but it is better to simplify it a bit. To do so, subtract the second square root from both sides and then square both sides to get:

Moving all the terms except for the square root from the right side to the left side and simplifying gives

Dividing both sides by 4 and squaring again gives

which can be re-arranged to get

Finally, since a is larger than c and both are positive, we can let b be a positive number such that . Substituting this into our last formula and dividing both sides by gives us the standard form of the equation of the ellipse:

The quantity e = c/a is called the eccentricity of the ellipse. The quantity b is called the semiminor axis of the ellipse. In particular, if the eccentricity of the ellipse is 0, then the two foci coincide, the semimajor and semiminor axes are equal, and the ellipse is a circle whose radius is precisely the semimajor axis.

Of course, one can also have ellipses with a vertical semimajor axis. If we simply interchange the roles of x and y in the above calculation, the final formula is the same, except that quantities a and b are interchanged. You distinguish the two cases by looking to see which of the two is the larger. You can also translate the graph of the equation to obtain ellipses centered at points (a, b) instead of at the origin (0, 0). As usual, the formulas look the same except that x is replaced with x - a and y is replace by y - b. Expansion and compression either horizontally or vertically just expands or contracts the values of a and b.

2.5.3 Hyperbolas

In analogy with Definition 5.2, we have

Definition 5.3: Let and be two not necessarily distinct points in the plane and let a be a positive real number. Then the ellipse with foci and and semimajor axis a is the set of points P = (x, y) such that the difference of the distances from P to the foci is exactly 2a. (Note that we have to subtract the smaller of two distances from the larger one in order to get 2a.)

Taking the foci (0, -c) and (0, c) as before, it is easy to use the distance formula to write down the equation of the hyperbola. You should go through steps just as with the ellipse. When you are done, you will see that the equation of the hyperbola reduces down to

where b is a positive number such that . As before, we call the eccentricity of the hyperbola. Note that e > 1 for hyperbolas and less than 1 for ellipses.

2.5.4 Reflective Properties of Conic Sections

2.5.4.1 Parabolas

Let be any point on the parabola . Clearly, the vertical line intersects the parabola in the single point . We will show that there is one other line that intersects the parabola in the single point ; this line is called the tangent line to the parabola at .

Proof: Any other line through P has an equation of the form . To find the points of intersection of this line with the parabola, one need only solve . This is a quadratic equation which can be rewritten as . The quadratic formula shows that this has only one solution precisely when . But this is the same as . Since we also have , it follows that there is only one solution precisely when But the expression on the left factors as and so there is only one solution if and only if m = x_1/(2d). We have shown that the equation of the tangent line to at is . Substituting , one gets the simpler expression:

for the tangent line. Clearly the y-intercept of the tangent line is exactly .

Proposition 5.1: Let be a point on the parabola . Then the tangent line to the parabola at the point has the equation

and so its y-intercept is at . Furthermore the angle between the vertical line and the tangent line is equal to the angle between the line through P and the focus F = (0, d) and the tangent line.

Proof: To show that the angles are equal, consider the triangle where is the point where the tangent line intersects the y-axis. The length of FQ is d + y_1 which is the same as the length of FP, because P is on the parabola and the distance from P to the directrix is y_1 + d. But, then the triangle is isosceles and so the base angles are equal: . Since the lines and the y-axis are parallel, it follows that the angle is equal to the angle between line and the tangent line.

2.5.4.2 Ellipses

Let P be any point on an ellipse E. Then a line L is said to be the tangent line to E at the point P if the line intersects E only at the single point P.

Proposition 5.2 The tangent line to the ellipse with equation at the point is the line with equation

Proof: One could proceed as in the case of parabolas by simply solving the system of equations consisting of the equation of the parabola and the arbitrary line through . However, the equations quickly become quite daunting.

Instead let's use polar coordinates. Consider the unit circle and the point on the circle. The slope of the line through the origin O and the point Q is clearly and so the slope of the line through Q perpendicular to this line has slope Its equation is Now, this is the tangent line to the circle at the point Q. If you don't know this from your geometry course, you can verify it via a calculation: Just substitute this value of into the equation of the unit circle. After you simplify the equation, you will have

and so the only solution is

Multiply the equation for the tangent line by and put only the constant term on the right side of the equation to get

Now, let's expand horizontally by a factor of a and vertically by a factor of b. This amounts to replacing the variables x and y in the formulas with x/a and y/b respectively. The equation of the unit circle transforms into the equation for the ellipse and the tangent line transforms into the tangent line

of the ellipse at the point Substituting these values into equation of the tangent line gives the equation as stated in the Proposition.

Proposition 5.3: Let P be a point on an ellipse E with foci and and L be the tangent line to E through the point P. The angle between the line and the tangent line is equal to the angle between the tangent line and the line .

Proof: This can be proved by a straightforward computation using the addition formula for tangents applied to the slopes of the various lines. However, there is a more illuminating proof.

Construct a line segment perpendicular to the tangent line and such that the distance from to the tangent line is equal to the distance from to the tangent line. Let H be the point on where the segment intersects the tangent line. Let Q be any point on the tangent line. We want to choose Q so that the sum of the distance from to Q and the distance from Q to G is least possible. Clearly, this happens when and are parts of the same line because the shortest distance between two points is a straight line. The triangle is isosceles and the tangent line cuts this triangle into two congruent pieces. One has . If and are parts of the same straight line, then is equal to the angle made by the tangent line and the line . So, the assertion would be proved if we just knew that this particular value of Q is, in fact, the point P.

To see that this is the point P, note that sum of the distance from to and the distance from Q to G is the same as the sum of the distance from to and the distance from Q to . Now Q was chosen so as to minimize the first sum, and so it also minimizes the second sum. So, if we knew that the second sum were minimized by Q = P, then we would know that our Q is P and the proof would be done.

Choose any point Q on the tangent line L. The line intersects the ellipse at a point R. Because R is on the ellipse, the sum of the distance from to R and the distance from R to must be 2a. Now, the distance from R to Q plus the distance from Q to must be greater or equal to the distance from R to with equality holding only if R is Q. So the distance from to Q plus the distance from Q to is larger than 2a whenever R is not Q. This means that the choice of Q which minimizes the sum of the distances is when R = Q, i.e. when Q is P. This completes the proof.

2.5.4.3 Hyperbolas

As you might expect, hyperbolas also have tangent lines reflective properties. A line L is tangent to a hyperbola H at a point on the hyperbola if it intersects the hyperbola at P and no other point.

Proposition 5.4: If is a point on the hyperbola , then the line

is tangent to the hyperbola at the point P.

Proof: Clearly, the equation is the equation of a line and the line passes through the point P. It remains to show that the line does not intersect the hyperbola in any other point. This is straightforward to check: solve the equation for the line for the variable and substitute into the equation of the hyperbola. Check to see the resulting equation in x has only one solution.

Proposition 5.5: Let P, H, and L be as in Proposition 5.4. Then the lines from the foci to the point P each make the same angle with the tangent line. One can also verify this by brute force: Suppose that and are both positive; the other cases being treated in a similar fashion. Let be such that the slope of L is , be such that the slope of the line from the left focus to P is and be such that the slope of the line from the right focus to P is . From a sketch, you can verify that the assertion of the proposition amounts to .

Calculating the slopes, one finds , , and . One can now substitute these values into the addition formula for tangents:

and

After simplifying algebraically and using , one sees that both of these quantities simplifies to which proves the result.

Remark: These reflective properties of conics have practical applications. For example, a simple satellite receiving antenna is in the shape of a parabola. You aim it at the satellite, i.e. align the axis of the parabola with the direction to the satellite, and so the signal is almost parallel to the axis; it bounces off the parabola and converges on the focus of the parabola, where you locate your receiver. A reflecting telescope works the same way; but there is a problem. You want the observer to be at the focus, but that is not possible because then the person would block the incoming light. To avoid this, you put a hyperolic reflector so that one of its foci is at the same location as the focus of the parabola and so that its other focus is at the vertex of the parabola. The light comes in parallel to the axis, bounces off the parabolic reflector headed towards the focus of the parabola, but hits the hyperbolic reflector and reflects off toward the focus at the vertex, where the observer sees the image.