Question 1

Which of the following is the correct center-radius form of the equation for a circle with center at $$(-3, 4)$$ and radius $$6$$?

Accepted Answer

$$(x + 3)^2$ + (y - 4)^2$ = 36$$

Question 2

Given the general form $$x^2$$ + $$y^2 + 10x - 4y + 33 = 0$, what is the correct conclusion about its graph?

Accepted Answer

The graph contains no points because the radius squared is negative

Question 3

What is the domain and range of the circle given by $$x^2$$ + $$y^2 = 9$?

Accepted Answer

Domain: $$[-3, 3]$$, Range: $$[-3, 3]$$

Question 4

A parabola has focus $$(0, 2)$$ and directrix $$y = -2. $$What is its equation?

Accepted Answer

$$x^2 = 8y$

Question 5

Which of the following best describes the set of all points $$(x, y)$$ that are equidistant from a fixed point (focus) and a fixed line (directrix)?

Accepted Answer

A parabola

Question 6

Given the equation $$x^2 - 6x$$ + $$y^2 + 10y + 25 = 0$, what are the center and radius of the circle after completing the square?

Accepted Answer

Center: $$(3, -5)$$, Radius: $$3$$

Question 7

A parabola with vertex at the origin and focus at $$(2/3, 0)$$ has which equation?

Accepted Answer

$$y^2$$ = \frac{8}{3}x$$

Question 8

The Parkes radio telescope has a parabolic cross-section with diameter $$210 ft $$and depth $$32 ft. $$Placing the vertex at the origin and opening upward, what is the value of $$c ($$the distance from vertex to focus)?

Accepted Answer

$$c = \frac{11,025}{128}$$

Question 9

Which of the following is the equation of a parabola with vertex $$(-1, 2)$$ and focus $$(2, 2)$$?

Accepted Answer

$$(y - 2)^2$ = 12(x + 1)$$

Question 10

Given the equation $$x = -2y^2$ + 4y - 3$$, what are the vertex and the direction in which the parabola opens?

Accepted Answer

Vertex: $$(-1, 1)$$, opens left

Question 11

Which of the following is the general form of a circle's equation?

Accepted Answer

$$x^2$$ + $$y^2 + Dx + Ey + F = 0$

Question 12

A parabola with equation $$(y + 3)^2$ = -8(x - 2)$$ has which vertex and focus?

Accepted Answer

Vertex: $$(2, -3)$$, Focus: $$(0, -3)$$

Question 13

If a parabola has the equation $$y^2 = -28x$, what is the location of its focus?

Accepted Answer

Focus: $$(-7, 0)$$

Question 14

Which of the following best describes the graph of $$x^2$$ + $$y^2 + Dx + Ey + F = 0$ when D$$, E, F$$ are real numbers?

Accepted Answer

It can be a circle, a point, or empty

Question 15

A parabola with vertex at the origin and passing through $$(-2, 12)$$ with a vertical axis has which equation?

Accepted Answer

$$x^2 = 4c y$ with c = 1/12$, so x^2$ = \frac{1}{3}y$$

Question 16

Which of the following is the correct equation for a circle with center at $$(\frac{3}{2}, -\frac{5}{2})$$ and radius $$3$$?

Accepted Answer

$$(x - \frac{3}{2})^2$ + (y + \frac{5}{2})^2$ = 9$$

Question 17

A real-world application: The Parkes radio telescope's parabolic cross-section is modeled by $$x^2$$ = \frac{11,025}{32}y$$. Where should the receiver be placed relative to the vertex?

Accepted Answer

At $$(0, 86.1)$$

Study Notes: Conic Sections – Circles and Parabolas (Precalculus, Chapter 8.1)

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