Math 170: Ideas in Mathematics

Homework 3

Due at recitation section, Feb 26-28, 2008


Conic sections

(1) Identify the type conic section described by each of the following equations in standard form. For each of them, give and graph the focus and directrix, and use these to then graph the conic section itself.

y = x2 + 2x + 3

(x/3)2+(y/2)2=1

(x/5)2-(y/6)2=1

(2) By computing Δ=B2-4AC, identify the type of conic section given by the quadratic equation in variables x, y

Ax2 + Bxy + Cy2 + Dx + Ey = 4x2 + 3xy + y2 - 4x + 2y = 0.

Bring it to standard form by the following variable changes. Set p = (BE-2CD)/Δ,  q = (DB-2AE)/Δ, and then plug x=X-p, y=Y-q into the equation and multiply it out. You should obtain a quadratic equation in X,Y with no linear terms.

Now make a second variable change by setting Y=y and X=(1/α)x-(B/2A)y where α=2 is the square root of A. Plug these new expressions for X,Y into your previous equation and multiply it out. The result should be an equation in the new variables x,y (not the same as the x, y above) in standard form.

Specify the type, focus and directrix. Sketch.

(3) Same question as number (2) for the quadratic equation

Ax2 + Bxy + Cy2 + Dx + Ey = x2 + xy - y2 - 5x + 5y = 0.
After the second substitution, either directly sketch a graph of the conic section you obtain, or, if you don't see how to get it into standard form, try switching x and y. What does this mean about the conic section you obtained?

(4) Consider an ellipse given in standard form by an equation

(x/a)2 + (y/b)2 = 1
with a > b. Sketch the ellipse, together with its foci given by the two points ( ± √ (a2-b2) , 0). The goal of this exercise is to prove that this ellipse satisfies the property most often referred to as the definition of an ellipse, namely that for every point (x,y) on the ellipse, the sum of distances to the two foci is a constant.

For this, we have to use the Pythagorean formula seen in class for the distance between a point P=(x,y) and a point F=(a,b>). This formula is given by

dist(P,F) = √ ( (x-a)2 + (y-b)2).

Use this formula to compute the distances from a point P=(x,y) on the ellipse to each of the two foci, and add them together. Use the substitution y2 = b2 - b2x2/a2 to simplify the expressions and prove that their sum is always equal to a constant.

(5) Apply question (4) to answer the following question. Suppose an ellipse E is given by the equation (x/3)2 + (y/2)2=1. We take a string, pin one end to a focus of E, and stretch it taut between that focus and any point of the ellipse, then from that point we stretch it taut to the other focus, pin it down and cut it off. What is the length of the string?

(6) Visualize a right circular cone in 3-space, cut by a vertical plane going straight down the middle, through the origin of 3-space. What is the conic section (the picture drawn on the vertical plane where the cone cuts it?)

(7) The equation (x/2)2+(y/2)2=1 looks like the standard form for the ellipse equation. But is it? Where are the foci? What is this curve?

(8) The properties of the parabola make it the ideal shape for the reflector of an automobile headlight. In the diagram below, the parabola is the shape of the reflecting metal back of the headlight, and the focus is where the light bulb is placed. What happens to the rays of light emanating from the light bulb that make this shape optimal for a headlight?

(9) Make a similar drawing for the ellipse with its foci. Imagine a speaker standing at one focus on an elliptic room, and sketch the rays of the sound of his voice reflecting off the walls. Where do they go? What effect does this have on the sound in the room?

(10) Describe the oldest (Greek, geometric) definition of the hyperbola. Describe an occurrence of the hyperbola in engineering or nature.