CONIC SECTIONS

Problem 1

For the equations below, identify each conic section. If it’s a parabola, specify its vertex, focus and directrix. If it’s an ellipse, specify its center, vertices and foci. If it’s a hyperbola, specify its center, vertices, foci and asymptotes.

a)

So the conic is an ELLIPSE, where , , and . We therefore conclude the following:

  • Its center is the point .
  • Its major axis is the horizontal line and its minor axis is the -axis.
  • Its vertices are located at the points and .
  • Its foci are located at the points and .

b)

So the conic is a PARABOLA that OPENS UP, where . We therefore conclude the following:

  • Its vertex is located at the point .
  • Its focus is located at the point .
  • Its directrix is the horizontal line .

c)

So the conic is a HYPERBOLA, where , , and . We therefore conclude the following:

  • Its center is the point .
  • Its transverse axis is the -axis and its conjugate axis is the vertical line .
  • Its vertices are located at the points and .
  • Its foci are located at the points and .
  • Its asymptotes are the lines and .

d)

So the conic is an ELLIPSE, where , , and . We therefore conclude the following:

  • Its center is the point .
  • Its major axis is the horizontal line and its minor axis is the vertical line .
  • Its vertices are located at the points and .
  • Its foci are located at the points and .

e)

So the conic is a HYPERBOLA, where , , and . We therefore conclude the following:

  • Its center is the point .
  • Its transverse axis is the vertical line and its conjugate axis is the horizontal line .
  • Its vertices are located at the points and .
  • Its foci are located at the points and .
  • Its asymptotes are the lines and .

Problem 2

Exercise # 72 on page 648: Reflecting Telescope

Since the mirror of the telescope is shaped like a paraboloid of revolution (i.e. a parabola rotated about its axis of symmetry), the collected light will be concentrated at the focus point of the parabola (or – technically – thecross-section of the paraboloid of revolution passing through its vertex).

Therefore, we are essentially looking for the y-coordinate of the focus point of a parabola that opens up, and has its vertex point at the origin of a Cartesian plane. Moreover, given the dimensions of the mirror given in the problem, we know that the pointsand belong to the parabola [do you see why?]

All of this implies that the equation of the parabola is given by , where , and that the focus is located at the point , orinches above the vertex point. Plugging in the coordinates of the point into the equation of the parabola yields, which in turn yields.

We conclude that the light will be concentrated exactly 0.1875 inches above the vertex point of the paraboloid of revolution.

Problem 3

Exercise # 78 on page 658: Volume of a Football

First, note that a longitudinal cross-section of the prolate spheroid (i.e. the football) passing through its center is an ellipse in the Cartesian plane whose center is at the origin, whose major axis is the x-axis, and whose vertices and foci lie on the x-axis.

Since the football is 11.125 inches in length, the vertices of the ellipse are located at the points and , and so we have.

Since the football has a center circumference of 28.25 inches, we have [do you see why?] This implies that .

We conclude that the volume of the football, in cubic inches, is given by

.

Problem 4

Exercise # 80 p. 672: Equilateral Hyperbola

We know that the eccentricity of an ellipse, denoted by , is defined as the positive number .

We also know that if the hyperbola is equilateral, then .

Since for any hyperbola, we conclude that for an equilateral hyperbola. Therefore its eccentricity is given by .

Problem 5

a)Identify the conic section represented by the polar equation .

Therefore, and the conic is a HYPERBOLA.

c)Find the rectangular equation of the conic. Justify your work analytically.

If , then we can convert the polar equation of the hyperbola to rectangular coordinates as follows:

PARAMETRIC EQUATIONS

Problem 1

Consider the plane curve defined by the following parametric equations:

,()

b)What type of graph do you recognize here?

A CARDIOID.

c)Find the polar equation of . Justify your work analytically.

If and, then we have

The polar equation of is therefore given by

.

Problem 2

Exercise # 54 p. 699 [projectile Motion]

Here we have , , and .

a)The parametric equations that model the position of the ball are given by

b)To find out how long the ball is in the air, find the positive root to the quadratic equation :

So the ball traveled for a total of 5.06 seconds.

c)To determine the horizontal distance that the ball traveled, compute :

So the ball traveled an approximate total horizontal distance of 484.52 feet.

d)To determine the maximal height attained by the ball, determine the y-coordinate of the vertex point of the parabola , or :

.

So the ball reached a maximal height of approximately 103.87 feet.