Question 1

Given points $$P(-5, -6)$$ and $$Q(7, -1)$$, what is the distance $$d(P, Q)$$ between them?

Accepted Answer

$$\sqrt{(7 + 5)^2$ + (-1 + 6)^2$} = \sqrt{12^2$ + 5^2$} = \sqrt{144 + 25} = \sqrt{169} = 13$$

Question 2

What are the coordinates of the midpoint $$M of $$the segment joining $$P(8, 2)$$ and $$Q(3, 5)$$?

Accepted Answer

$$\left( \frac{8 + 3}{2}, \frac{2 + 5}{2} \right) = (5.5, 3.5)$$

Question 3

Are the points $$(-6, -4)$$, $$(0, -2)$$, and $$(-10, 8)$$ the vertices of a right triangle?

Accepted Answer

Yes, because the squares of the distances satisfy the Pythagorean theorem.

Question 4

Given points $$(0, -7)$$, $$(-3, 5)$$, and $$(2, -15)$$, are these points collinear?

Accepted Answer

No, because the slopes between each pair are not equal.

Question 5

A vertical line is drawn through the point $$(4, 3). At $$what point does it intersect the $$x-$$axis?

Accepted Answer

$$(4, 0)$$

Question 6

A horizontal line is drawn through the point $$(4, 3). At $$what point does it intersect the $$y-$$axis?

Accepted Answer

$$(0, 3)$$

Question 7

Which of the following equations represents a circle, and what is its center and radius? $$x^2$$ + $$y^2 + 6x + 8y + 9 = 0$

Accepted Answer

Yes, center $$(-3, -4)$$, radius $$2$$

Question 8

Find the equation of the circle with center $$(-4, 3)$$ passing through the point $$(5, 8) in $$center-radius form.

Accepted Answer

$$(x + 4)^2$ + (y - 3)^2$ = 106$$

Question 9

Find all points $$(x, y)$$ with $$x = y$$ that are $$4$$ units from $$(1, 3). $$Which of the following is a correct solution?

Accepted Answer

The points $$(\frac{1 + 3}{2} + 2, \frac{1 + 3}{2} + 2)$$ and $$(\frac{1 + 3}{2} - 2, \frac{1 + 3}{2} - 2)$$

Question 10

Find all points satisfying $$x + y = 0$$ that are $$8$$ units from $$(-2, 3). $$Which of the following is a correct solution?

Accepted Answer

The points $$(x, -x)$$ where $$\sqrt{(x + 2)^2$ + (-x - 3)^2$} = 8$$

Question 11

Which of the following equations does NOT represent a circle?

Accepted Answer

Question 12

Given the equation $$9x^2$$ + $$9y^2 - 6x + 6y - 23 = 0$, what is the center and radius of the circle?

Accepted Answer

Center $$(\frac{1}{3}, -\frac{1}{3})$$, radius $$\sqrt{\frac{26}{9}}$$

Question 13

Find the coordinates of all points whose distance from $$(1, 0) is$$ $$\sqrt{10}$$ and whose distance from $$(5, 4) is$$ $$\sqrt{10}. $$Which of the following is correct?

Accepted Answer

The points $$(3, 2)$$ and $$(-1, -2)$$

Question 14

A chemical reaction produces a circular spill on a lab table. If the spill's center is at $$(2, -3)$$ and it touches the point $$(6, 1)$$, what is the equation of the spill's boundary?

Accepted Answer

$$(x - 2)^2$ + (y + 3)^2$ = 32$$

Question 15

A medical imaging device scans a circular region with center $$(0, 0)$$ and radius $$5. $$Which equation describes the scanned region?

Accepted Answer

$$x^2$$ + $$y^2 = 25$

Question 16

A city plans to build a circular park. The center is at $$(4, 7)$$, and the park must pass through $$(10, 7). $$What is the equation of the park's boundary?

Accepted Answer

$$(x - 4)^2$ + (y - 7)^2$ = 36$$

Question 17

A horizontal line passes through $$(5, -2). At $$what point does it intersect the $$y-$$axis?

Accepted Answer

$$(0, -2)$$

Question 18

A vertical line passes through $$(-3, 7). At $$what point does it intersect the $$x-$$axis?

Accepted Answer

$$(-3, 0)$$

College Algebra: Coordinate Geometry and Circles Study Guide

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