Question 1

Which of the following is the standard form equation of an ellipse centered at the origin with a horizontal major axis, where the vertices are at $$(\pm a, 0)$$ and the co-vertices are at $$(0, \pm b)$$?

Accepted Answer

$$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$

Question 2

Given the parabola $$y^2 = 8x$, what are the coordinates of the focus?

Accepted Answer

$$(2, 0)$$

Question 3

Which of the following equations represents a hyperbola centered at the origin with foci on the x-axis?

Accepted Answer

$$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$

Question 4

If the equation of a circle is $$(x-3)^2$$ + $$(y+2)^2 = 16$, what are the center and radius of the circle?

Accepted Answer

Center: $$(3, -2)$$, Radius: $$4$$

Question 5

Which of the following is the equation of the directrix for the parabola $$y^2 = 4px$?

Accepted Answer

$$x = -p$$

Question 6

A satellite dish is shaped like a paraboloid. If the receiver is placed at the focus and the equation of the cross-section is $$y^2 = 12x$, how far from the vertex should the receiver be placed?

Accepted Answer

$$3$$ units

Question 7

Given the ellipse $$\frac{x^2$}{9} + \frac{y^2$}{4} = 1$$, what is the length of the major axis?

Accepted Answer

$$6$$

Question 8

Which of the following best describes the graph of $$x^2$$ + $$y^2 - 6x + 8y + 9 = 0$?

Accepted Answer

A circle with center $$(3, -4)$$ and radius $$2$$

Question 9

For the hyperbola $$\frac{y^2$}{16} - \frac{x^2$}{9} = 1$$, what are the coordinates of the vertices?

Accepted Answer

$$(0, 4)$$ and $$(0, -4)$$

Question 10

A real-world application: The path of a thrown ball can be modeled by a parabola. If the equation is $$y = -2x^2$ + 8x + 3$$, what is the maximum height the ball reaches?

Accepted Answer

$$11$$

Question 11

Which of the following is the correct formula for the distance between the foci of the ellipse $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$ where $$a \u003e b$$?

Accepted Answer

$$2\sqrt{a^2$ - b^2$}$$

Question 12

Given the equation $$x^2$$ - $$4y^2 = 16$, which type of conic section does this represent?

Accepted Answer

Hyperbola

Question 13

If a circle passes through the points $$(1,2)$$ and $$(5,2)$$ and has its center on the x-axis, what is the equation of the circle?

Accepted Answer

$$(x-3)^2$$ + $$y^2 = 10$

Question 14

Which of the following is the equation of a parabola that opens upward with vertex at $$(0,0)$$ and focus at $$(0,2)$$?

Accepted Answer

$$x^2 = 8y$

Question 15

A bridge is designed in the shape of a semi-ellipse with a span of $$40$$ meters and a maximum height of $$10$$ meters. What is the equation of the ellipse if the center is at the origin?

Accepted Answer

$$\frac{x^2$}{400} + \frac{y^2$}{100} = 1$$

College Algebra Review: Circles, Parabolas, Ellipses, and Matrices

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