Question 1

Which of the following is the correct formula for the distance $$d$$ between two points $$A(x_1$, y_1$)$$ and $$B(x_2$, y_2$) in $$the coordinate plane?

Accepted Answer

$$d = \sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$

Question 2

Given the points $$A(-1, -3)$$ and $$B(2, 3)$$, what is the distance between them? Express your answer in simplified radical form.

Accepted Answer

$$\sqrt{(2 - (-1))^2$ + (3 - (-3))^2$} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5}$$

Question 3

What is the midpoint of the line segment with endpoints $$(1, 2)$$ and $$(7, -3)$$?

Accepted Answer

$$(4, -0.5)$$

Question 4

Which formula correctly gives the midpoint $$M of a $$segment with endpoints $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

$$M = \left( \dfrac{x_1$ + x_2$}{2}, \dfrac{y_1$ + y_2$}{2} \right)$$

Question 5

A circle has center at $$(0, -6)$$ and radius $$10. $$What is its equation in standard form?

Accepted Answer

$$(x - 0)^2$ + (y + 6)^2$ = 100$$

Question 6

Given the equation of a circle $$x^2$$ + $$y^2 + 4x - 6y = 12$, what are the center and radius of the circle?

Accepted Answer

Center $$(-2, 3)$$, radius $$\sqrt{25} = 5$$

Question 7

Which of the following is the general form of the equation of a circle?

Accepted Answer

$$x^2$$ + $$y^2 + Dx + Ey + F = 0$

Question 8

Convert the equation $$x^2$$ + $$y^2 + 4x - 6y = 12$ to standard form.

Accepted Answer

$$(x + 2)^2$ + (y - 3)^2$ = 25$$

Question 9

A cell phone tower is located at $$(2, 5). A $$city wants to build a circular park with the tower at the center and a radius of $$7$$ units. What is the equation of the park's boundary?

Accepted Answer

$$(x - 2)^2$ + (y - 5)^2$ = 49$$

Question 10

A circle has the equation $$(x - 3)^2$ + (y + 1)^2$ = 16. $$Which of the following statements is true?

Accepted Answer

The center is $$(3, -1)$$ and the radius is $$4$$

Question 11

Given the endpoints of a diameter of a circle at $$(2, 8)$$ and $$(10, 4)$$, what is the equation of the circle?

Accepted Answer

$$(x - 6)^2$ + (y - 6)^2$ = 20$$

Question 12

A circle has the equation $$x^2$$ + $$y^2 - 8x + 6y + 9 = 0$. What is the center and radius of the circle?

Accepted Answer

Center $$(4, -3)$$, radius $$4$$

Question 13

A city plans to build a circular fountain such that its center is at $$(5, -2)$$ and it passes through the point $$(9, 1). $$What is the equation of the fountain's boundary?

Accepted Answer

$$(x - 5)^2$ + (y + 2)^2$ = 25$$

Question 14

Which of the following best describes the relationship between the standard and general forms of a circle's equation?

Accepted Answer

The standard form can be converted to general form by expanding and simplifying.

Question 15

A circle has the equation $$x^2$$ + $$y^2 = 49$. What are the center and radius?

Accepted Answer

Center $$(0, 0)$$, radius $$7$$

Question 16

If the distance between two points is $$10$$ and one point is at $$(3, 4)$$, which of the following could be the coordinates of the other point?

Accepted Answer

$$(3, 14)$$

Question 17

A circle has the equation $$x^2$$ + $$y^2 - 2x + 4y = 20$. What is the center and radius?

Accepted Answer

Center $$(1, -2)$$, radius $$5$$

Question 18

A circle passes through the points $$(0, 0)$$ and $$(6, 0)$$ and has its center on the $$x-$$axis. What is the equation of the circle?

Accepted Answer

$$(x - 3)^2$ + y^2$ = 9$$

Question 19

A circle has the equation $$(x + 1)^2$ + (y - 2)^2$ = 13. $$Which of the following points lies on the circle?

Accepted Answer

$$(2, 5)$$

Question 20

A circle has center $$(h, k)$$ and passes through the point $$(h + 5, k + 12). $$What is the radius of the circle?

Accepted Answer

$$13$$

Distance and Midpoint Formulas; Circles

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