Q: For the ellipse $$\frac{x^2$}{25} + \frac{y^2$}{16} = 1$$, what are the coordinates of the foci?

$$(\pm 3, 0)$$,\text{ where } c = \sqrt{$$a^2$$ - $$b^2$$} = 3$

Question 1

Which of the following best describes the geometric definition of a parabola?

Accepted Answer

A parabola is the set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix).

Question 2

Given the equation $$y^2 = 4ax$, what is the location of the focus and the equation of the directrix?

Accepted Answer

Focus: $$(a, 0)$$; Directrix: $$x = -a$$

Question 3

What is the length of the latus rectum for the parabola $$y^2 = 4ax$?

Accepted Answer

$$4a$$

Question 4

A satellite dish is shaped like a paraboloid of revolution. If the dish is 8 feet across at its opening and 2 feet deep at its center, at what position should the receiver be placed?

Accepted Answer

1 foot above the vertex

Question 5

Which of the following is the standard form of the equation of an ellipse with center at the origin, major axis along the x-axis, and lengths $$2a$$ and $$2b$$?

Accepted Answer

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

Question 6

For the ellipse $$\frac{x^2$}{25} + \frac{y^2$}{16} = 1$$, what are the coordinates of the foci?

Accepted Answer

$$(\pm 3, 0)$$,	ext{ where } c = \sqrt{$$a^2$$ - $$b^2$$} = 3$

Question 7

Which of the following equations represents a hyperbola with center at the origin and transverse axis along the y-axis?

Accepted Answer

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

Question 8

What are the equations of the asymptotes for the hyperbola $$\frac{x^2$}{a^2$} - \frac{y^2$}{b^2$} = 1$$?

Accepted Answer

$$y = \pm \frac{b}{a}x$$

Question 9

Given the equation $$4x^2$$ - $$9y^2 = 36$, what type of conic section does this represent and what are its asymptotes?

Accepted Answer

Hyperbola; $$y = \pm \frac{2}{3}x$$

Question 10

Which of the following is the partial fraction decomposition of $$\frac{4}{x^2$ - x}$$?

Accepted Answer

$$\frac{4}{x} - \frac{4}{x-1}$$

Question 11

For the ellipse $$16x^2$$ + $$y^2 = 16$, what are the lengths of the major and minor axes?

Accepted Answer

Major axis: $$4$$, Minor axis: $$1$$

Question 12

Which of the following is the correct partial fraction decomposition for $$\frac{x^2$ - 4x + 12}{x^2$ - 4x^2$ + 4x}$$?

Accepted Answer

$$\frac{A}{x} + \frac{B}{x-2} + \frac{C}{(x-2)^2$}$$

Question 13

A point C(x) represents the location of a lightning strike. Sound travels at 1100 feet per second, so the person at point A is 1100 feet from the source. If the person at point B is 3300 feet from the source, the locus of points C(x) is:

Accepted Answer

A hyperbola with foci at A and B and difference of distances 2200

Question 14

Which of the following is the standard form of the equation of a hyperbola with center at $$(h, k)$$ and transverse axis parallel to the x-axis?

Accepted Answer

$$\frac{(x-h)^2$}{a^2$} - \frac{(y-k)^2$}{b^2$} = 1$$

Question 15

If the equation of a parabola is $$x^2 = 8y$, what is the vertex, focus, and directrix?

Accepted Answer

Vertex: $$(0, 0)$$; Focus: $$(0, 2)$$; Directrix: $$y = -2$$

Question 16

Which of the following is the correct partial fraction decomposition for $$\frac{2x^2$ + x - 5}{x^2$ + 1}$$?

Accepted Answer

$$2x + 1 + \frac{-5}{x^2$ + 1}$$

Question 17

The equation $$x^2 = 4ay$ represents a parabola that opens:

Accepted Answer

Upward

Question 18

For the ellipse $$\frac{x^2$}{a^2$} + \frac{y^2$}{b^2$} = 1$$ with $$a \u003e b$$, what is the distance from the center to each focus?

Accepted Answer

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

Question 19

Which of the following is the correct partial fraction decomposition for $$\frac{2x^2$ + x - 5}{(x^2$ + 1)^2$}$$?

Accepted Answer

$$\frac{Ax + B}{x^2$ + 1} + \frac{Cx + D}{(x^2$ + 1)^2$}$$

Conic Sections and Partial Fraction Decomposition: Study Notes for Precalculus

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