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?

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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