Before you read

Answer first — see what you already know.

A radio tower broadcasts in a circular range described by $(x - 4)^{2} + (y + 1)^{2} = 36$ . What is the center and broadcast radius?

A
Center $(4, - 1)$ , radius $6$
B
Center $(- 4, 1)$ , radius $6$
C
Center $(4, - 1)$ , radius $36$
D
Center $(4, 1)$ , radius $6$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A sprinkler is placed at coordinates $(0, 5)$ and has a spray radius of 3 meters. Write the equation of the coverage area.

A
$x^{2} + (y - 5)^{2} = 3$
B
$x^{2} + (y - 5)^{2} = 9$
C
$(x - 5)^{2} + y^{2} = 9$
D
$x^{2} + y^{2} = 25$

Check your answer

Answer:

Theory

Standard Form of a Circle

The equation of a circle with center $(h, k)$ and radius $r$ :
$(x - h)^2 + (y - k)^2 = r^2$

Circle with center (h,k) and radius r

Examples:
- Center $(3, -2)$ , radius $5$ : $(x-3)^2 + (y+2)^2 = 25$
- Center at origin, radius $4$ : $x^2 + y^2 = 16$

General form: $x^2 + y^2 + Dx + Ey + F = 0$ . To find center and radius, complete the square.

Example 1

Write the equation: $x^2 + y^2 - 6x + 4y - 12 = 0$ in standard form.

Group: $(x^2 - 6x) + (y^2 + 4y) = 12$

Complete the square: $(x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4$

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

Center: $(3, -2)$ , Radius: $5$

Center

(3, -2)

, radius

5

Strategy

SAT Pro Tip

To complete the square for $x^2 + bx$ : take half of $b$ , square it, and add it to both sides. For $x^2 - 6x$ : half of $-6$ is $-3$ , squared is $9$ . So add $9$ .

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