Exercises: Equation of a circle (standard form, completing the square) - Digital SAT Math

A

Recall / Warm-Up

1.

Which of the following is the correct expansion of $(a + b)^2$ ?

A.

$a^2 + b^2$

B.

$a^2 + 2ab + b^2$

C.

$a^2 - 2ab + b^2$

D.

$2a + 2b$

2.

The distance formula gives the distance between $(1, 2)$ and $(4, 6)$ as
$d = \\sqrt{(4-1)^2 + (6-2)^2}$ .
Compute $d$ .

3.

To complete the square on $x^2 - 8x$ : take half of $-8$ , which is ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ , then square it to get ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ .
Add this to get $(x - \\_\\_\\_)^2$ .

half coefficient:

square of half:

value in factored form:

B

Fluency Practice

1.

A circle has center $(2, -5)$ and radius $3$ .
What is the value of $r^2$ in its standard-form equation $(x - 2)^2 + (y + 5)^2 = r^2$ ?

2.

A circle has center $(-3, 4)$ and passes through the origin $(0, 0)$ .
What is the radius of the circle?

3.

What is the center of the circle $(x + 1)^2 + (y - 7)^2 = 36$ ?

A.

$(1, 7)$

B.

$(1, -7)$

C.

$(-1, 7)$

D.

$(-1, -7)$

4.

What is the radius of the circle $(x - 4)^2 + (y + 3)^2 = 49$ ?

A.

$49$

B.

$24.5$

C.

$7$

D.

$\\sqrt{24.5}$

5.

A circle has the equation $x^2 + y^2 = 25$ .
What are the coordinates of the center? Enter your answer as an ordered pair (e.g., $(0, 0)$ ).

C

Varied Practice

1.

Which equation represents a circle with center $(-2, 3)$ and radius $4$ ?

A.

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

B.

$(x - 2)^2 + (y + 3)^2 = 16$

C.

$(x + 2)^2 + (y - 3)^2 = 16$

D.

$(x - 2)^2 + (y - 3)^2 = 16$

2.

The equation of a circle is $(x - 5)^2 + (y + 2)^2 = 81$ .
What is the diameter of the circle?

3.

Which statement correctly describes the circle $(x + 3)^2 + (y - 1)^2 = 25$ ?

A.

Center $(3, -1)$ , radius $5$

B.

Center $(-3, 1)$ , radius $25$

C.

Center $(-3, 1)$ , radius $5$

D.

Center $(3, 1)$ , radius $5$

4.

Convert the equation $x^2 + y^2 - 6x + 4y - 12 = 0$ to standard form by completing the square.
What is the radius of the circle?

5.

Convert $x^2 + y^2 + 10x - 2y + 22 = 0$ to standard form.
What is the $x$ -coordinate of the center?

D

Word Problems / Application

1.

A circle has its center at $(1, -3)$ and passes through the point $(1, 2)$ .

1.

What is the radius of the circle?

2.

Write the equation of the circle in standard form. What is the value on the right side ( $r^2$ )?

2.

The endpoints of a diameter of a circle are $A(2, 4)$ and $B(8, 4)$ .

What is the radius of this circle?

3.

A circle passes through the three points $(0, 0)$ , $(6, 0)$ , and $(0, 8)$ .

Which equation represents this circle?

A.

$(x - 3)^2 + (y - 4)^2 = 25$

B.

$x^2 + y^2 = 100$

C.

$(x - 6)^2 + (y - 8)^2 = 100$

D.

$(x - 3)^2 + (y - 4)^2 = 100$

E

Error Analysis

1.

Alex identified the center and radius of $(x + 4)^2 + (y - 2)^2 = 9$ as follows:

Center: $(4, -2)$
Radius: $9$

Alex made two errors. Which statement correctly identifies both errors?

A.

The center is correct but the radius should be $3$ .

B.

The center should be $(-4, 2)$ because standard form uses subtraction, so $(x+4)^2 = (x-(-4))^2$ means $h = -4$ ; Alex also forgot to take the square root for the radius, so $r = \\sqrt{9} = 3$ .

C.

The center is correct; only the radius is wrong ( $r = 3$ , not $9$ ).

D.

Both the center and radius are correct.

2.

Jordan converted $x^2 + y^2 - 4x + 6y + 4 = 0$ as follows:

Step 1: $(x^2 - 4x + 4) + (y^2 + 6y + 9) = 4$

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

What error did Jordan make in the conversion?

A.

Jordan grouped the terms incorrectly.

B.

Jordan computed the completing-the-square constants incorrectly.

C.

Jordan added $4$ and $9$ to the left side but did not add them to the right side — the right side should be $4 + 4 + 9 = 17$ , giving $(x - 2)^2 + (y + 3)^2 = 17$ .

D.

Jordan's work is correct.

F

Challenge / Extension

1.

The equation $x^2 + y^2 + 6x - 8y + k = 0$ represents a circle of radius $2$ .
Find the value of $k$ .

2.