Q: Which of the following is the correct polar form of the equation $$x^2 = 10y$?

$$r^2$$ \$$cos^2$$ \theta = 10r \sin \theta$$

Q: Transform the polar equation $$r = -2 \cos \theta to $$rectangular coordinates and identify the graph.

$$x^2$$ + $$y^2 + 2x = 0$, \text{circle with center at } (-1, 0)$

Q: Find the equation of the parabola with vertex at $$(0,0)$$ and focus at $$(-1,0). $$What are the coordinates of the endpoints of the latus rectum?

$$y^2 = -4x$, \text{endpoints: } ( -1, 2 ), ( -1, -2 )$

Q: Given the ellipse $$\frac{x^2$}{25} + \frac{y^2$}{9} = 1$$, what is the value of $$a$$, the value of $$b$$, and the major axis?

$$a = 5$$, $$b = 3$$, \text{major axis is the } x\text{-axis}$

Question 1

Convert the polar coordinates $$\left(5, \frac{2\pi}{3}\right) to $$rectangular coordinates.

Accepted Answer

$$\left(-\frac{5}{2}, \frac{5\sqrt{3}}{2}\right)$$

Question 2

Given the rectangular coordinates $$(-10, 0)$$, what are the polar coordinates with $$r \u003e 0$$ and $$0 \leq 	heta \u003c 2\pi$$?

Accepted Answer

$$(10, \pi)$$

Question 3

Which of the following is the correct polar form of the equation $$x^2 = 10y$?

Accepted Answer

$$r^2$$ \$$cos^2$$ 	heta = 10r \sin 	heta$$

Question 4

Transform the polar equation $$r = -2 \cos 	heta to $$rectangular coordinates and identify the graph.

Accepted Answer

$$x^2$$ + $$y^2 + 2x = 0$, 	ext{circle with center at } (-1, 0)$

Question 5

Find the magnitude of the vector $$\vec{v} = -6\hat{i} + 8\hat{j}.$$

Accepted Answer

$$10$$

Question 6

Given $$\vec{v} = 3\hat{i} - 4\hat{j}$$, what is the direction angle of $$\vec{v} (in $$degrees, rounded to the nearest integer)?

Accepted Answer

$$307^\circ$$

Question 7

Given $$\vec{v} = 5\hat{i} + 12\hat{j}$$ and $$\vec{w} = 5\hat{i} - 12\hat{j}$$, what is the angle between $$\vec{v}$$ and $$\vec{w}$$?

Accepted Answer

$$90^\circ$$

Question 8

Find the equation of the parabola with vertex at $$(0,0)$$ and focus at $$(-1,0). $$What are the coordinates of the endpoints of the latus rectum?

Accepted Answer

$$y^2 = -4x$, 	ext{endpoints: } ( -1, 2 ), ( -1, -2 )$

Question 9

A satellite dish is shaped like a paraboloid of revolution. If the dish is 16 feet across at its opening and 4 feet deep at its center, at what position should the receiver be placed from the base of the dish, along its axis of symmetry?

Accepted Answer

$$1$$ foot

Question 10

Given the ellipse $$\frac{y^2$}{16} + \frac{x^2$}{81} = 1$$, what are the equations of the asymptotes for the related hyperbola $$\frac{y^2$}{16} - \frac{x^2$}{81} = 1$$?

Accepted Answer

$$y = \frac{9}{4}x$$ and $$y = -\frac{9}{4}x$$

Question 11

Find the polar coordinates for the point $$(5, 5\sqrt{3})$$ with $$r \u003e 0$$ and $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

$$\left(10, \frac{\pi}{3}\right)$$

Question 12

Given the vectors $$\vec{v} = \hat{i} + \hat{j}$$ and $$\vec{w} = -\hat{i} + \hat{j}$$, what is their dot product and are they orthogonal, parallel, or neither?

Accepted Answer

Dot product is $$0$$, and the vectors are orthogonal.

Question 13

A mirror is shaped like a paraboloid of revolution. If the mirror is 30 feet across at its opening and 11 feet deep, where will the heat source be concentrated above the bottom of the mirror?

Accepted Answer

$$\frac{225}{44}$$ feet

Question 14

Given the ellipse $$\frac{x^2$}{25} + \frac{y^2$}{9} = 1$$, what is the value of $$a$$, the value of $$b$$, and the major axis?

Accepted Answer

$$a = 5$$, $$b = 3$$, 	ext{major axis is the } x	ext{-axis}$

Question 15

Find the equation of the ellipse with center at $$(0,0)$$, focus at $$(0,12)$$, and vertex at $$(0,13).$$

Accepted Answer

$$\frac{x^2$}{25} + \frac{y^2$}{169} = 1$$

Question 16

Given the hyperbola $$\frac{y^2$}{16} - \frac{x^2$}{81} = 1$$, what are the equations of its asymptotes?

Accepted Answer

$$y = \frac{9}{4}x$$ and $$y = -\frac{9}{4}x$$

Question 17

Given the vectors $$\vec{v} = 9\hat{i}$$ and $$\vec{w} = -9\hat{j}$$, what is the angle between them?

Accepted Answer

$$90^\circ$$

Question 18

Given the equation $$	heta = \frac{7\pi}{4} in $$polar coordinates, what is the slope-intercept form in rectangular coordinates?

Accepted Answer

$$y = -x$$

Precalculus Trigonometry Review: Polar Coordinates, Vectors, Conic Sections

Study Guide - Practice Questions

Study Guide - Flashcards