The equation x^2 + y^2 - 14x - 8y + 40 = 0 defines a circle in the xy-plane. If the center… — SAT

The equation $x^2 + y^2 - 14x - 8y + 40 = 0$ defines a circle in the $xy$ -plane. If the center of the circle has coordinates $(h, k)$ , what is the value of $h + k$ ?

Answer: 11

Answer

The correct answer is 11.

The equation is rewritten in the standard form

(x - 7)^2 + (y - 4)^2 = 25

by completing the square. The center coordinates

(h, k)

are

(7, 4)

. Thus,

h + k = 7 + 4 = 11

Step-by-Step Solution

Group terms and complete the square for x and y

The equation becomes

(x - 7)^2 + (y - 4)^2 = 25

To convert the general form of the circle's equation into the standard form

(x - h)^2 + (y - k)^2 = r^2

Identify the coordinates of the center

The center

(h, k)

(7, 4)

, so

h = 7

and

k = 4

The standard form directly gives the center coordinates as

(h, k)

when written as

(x - h)^2 + (y - k)^2 = r^2

Calculate the sum of the coordinates

h + k = 7 + 4 = 11

The question asks for the value of

h + k

Key Concept

Completing the square to find the standard form of a circle's equation and identifying the center coordinates.

Question

Answer

Step-by-Step Solution

Key Concept

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