In the xy-plane, a circle with center (4, 9) and radius 5 is defined by the equation (x - … — SAT

In the $xy$ -plane, a circle with center $(4, 9)$ and radius $5$ is defined by the equation $(x - h)^2 + (y - 9)^2 = 25$ , where $h$ is a positive constant. What is the value of $h$ ?

Answer: 4

Answer

The value of the constant

h

4

The standard equation of a circle with center

(h, k)

and radius

r

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

. Since the center is

(4, 9)

and the radius is

5

, the equation of the circle is

(x - 4)^2 + (y - 9)^2 = 5^2

, which simplifies to

(x - 4)^2 + (y - 9)^2 = 25

. Comparing this to the given equation

(x - h)^2 + (y - 9)^2 = 25

, we see that the constant

h

corresponds to the

x

-coordinate of the center, which is

4

Step-by-Step Solution

Identify the standard equation of a circle.

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

This formula represents a circle with center

(h, k)

and radius

r

in the coordinate plane.

Substitute the given center coordinates into the standard equation.

For center

(4, 9)

, the equation is

(x - 4)^2 + (y - 9)^2 = r^2

The coordinates of the center are mapped directly to

h

and

k

in the standard form.

Compare the equation template to find the value of

h

The equation

(x - 4)^2 + (y - 9)^2 = 25

matches the format

(x - h)^2 + (y - 9)^2 = 25

, indicating that

h = 4

Equating the terms in both equations allows us to identify the value of the positive constant

h

Key Concept

Extracting coordinates of the center from the standard form equation of a circle.

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