Question 1

Given the equation $$4x^2$$ - $$9y^2 - 9x + 36y - 68 = 0$, what is the center of the hyperbola after completing the square?

Accepted Answer

$$(1, 2)$$

Question 2

Which of the following is the standard form of the hyperbola $$4x^2$$ - $$9y^2 - 9x + 36y - 68 = 0$ after completing the square?

Accepted Answer

$$\frac{(x-1)^2$}{9} - \frac{(y-2)^2$}{4} = 1$$

Question 3

For the hyperbola $$\frac{(x-1)^2$}{9} - \frac{(y-2)^2$}{4} = 1$$, what are the lengths of the transverse and conjugate axes?

Accepted Answer

Transverse axis: $$2a = 6$$, Conjugate axis: $$2b = 4$$

Question 4

Which of the following equations represents a hyperbola with center $$(2, -2)$$, $$a = 4$$, and $$b = 3$$?

Accepted Answer

$$\frac{(x-2)^2$}{16} - \frac{(y+2)^2$}{9} = 1$$

Question 5

Given the hyperbola $$\frac{(y+4)^2$}{25} - \frac{(x+2)^2$}{9} = 1$$, what is the orientation of its transverse axis?

Accepted Answer

Vertical

Question 6

What are the equations of the asymptotes for the hyperbola $$\frac{(x-1)^2$}{9} - \frac{(y-2)^2$}{4} = 1$$?

Accepted Answer

$$y - 2 = \pm \frac{2}{3}(x - 1)$$

Question 7

A real-world application: A satellite dish is shaped like a parabola. If its equation is $$y = \frac{1}{2}(x-3)^2$ + 2$$, what is the vertex of the dish?

Accepted Answer

$$(3, 2)$$

Question 8

Given the equation $$x^2 + 4x - 5y - 11 = 0$, what is the vertex of the parabola after completing the square?

Accepted Answer

$$(2, 3)$$

Question 9

For the parabola $$y^2 + 2y = -2x - 7$, what is the direction in which it opens?

Accepted Answer

Left

Question 10

Which of the following is the equation of a parabola with vertex $$(3, 2)$$ that opens upward?

Accepted Answer

$$y = \frac{1}{2}(x-3)^2$ + 2$$

Question 11

A medical imaging device uses a hyperbolic mirror described by $$\frac{(y+4)^2$}{25} - \frac{(x+2)^2$}{9} = 1. $$What are the coordinates of its center?

Accepted Answer

$$( -2, -4 )$$

Question 12

For the hyperbola $$\frac{(x-2)^2$}{16} - \frac{(y+2)^2$}{9} = 1$$, what is the distance between its foci?

Accepted Answer

$$2c = 2\sqrt{a^2$ + b^2$} = 2\sqrt{16 + 9} = 2\sqrt{25} = 10$$

Question 13

Which conic section is represented by the equation $$x^2 + 4x - 5y - 11 = 0$?

Accepted Answer

Parabola

Question 14

A bridge arch is modeled by the equation $$y = -\frac{1}{2}(x+1)^2$ + 3. $$What is the maximum height of the arch?

Accepted Answer

$$3$$

Question 15

Which of the following is the correct standard form for a hyperbola with center $$(2, -4)$$, $$a = 3$$, and $$b = 5$$, opening vertically?

Accepted Answer

$$\frac{(y+4)^2$}{25} - \frac{(x+2)^2$}{9} = 1$$

Question 16

If a hyperbola has the equation $$\frac{(x-h)^2$}{a^2$} - \frac{(y-k)^2$}{b^2$} = 1$$, which variable determines the orientation of the transverse axis?

Accepted Answer

$$a$$

Question 17

A parabolic reflector is described by $$y = \frac{1}{2}(x-3)^2$ + 2. If $$the reflector is shifted 2 units up and 1 unit right, what is the new equation?

Accepted Answer

$$y = \frac{1}{2}(x-4)^2$ + 4$$

Question 18

Which of the following best describes the asymptotes of a hyperbola in the form $$\frac{(x-h)^2$}{a^2$} - \frac{(y-k)^2$}{b^2$} = 1$$?

Accepted Answer

$$y - k = \pm \frac{b}{a}(x - h)$$

Analytic Geometry: Hyperbolas and Parabolas

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