$(x + 4)^2 + (y - 19)^2 = 121$ $\newline$ The graph of the given equation is a circle in the $xy$ -plane. The point $(a, b)$ lies on the circle. Which of the following is a possible value for $a$ ? $\newline$ A) $-16$ $\newline$ B) $-14$ $\newline$ C) $11$ $\newline$ D) $19$

Full solution

(x + 4)^2 + (y - 19)^2 = 121

\newline

The graph of the given equation is a circle in the

xy

-plane. The point

(a, b)

lies on the circle. Which of the following is a possible value for

a

\newline

-16

\newline

-14

\newline

11

\newline

19

Circle Equation Analysis: The equation of the circle is given by $(x + 4)^2 + (y - 19)^2 = 121$ . To find a possible value for $a$ , we need to understand that the center of the circle is at $(-4, 19)$ and the radius is the square root of $121$ , which is $11$ . Any point $(a, b)$ that lies on the circle must satisfy the equation of the circle.
Finding Possible Value for $a$ : We can choose a value for $a$ that is $11$ units away from the x-coordinate of the center of the circle since the radius is $11$ . The x-coordinate of the center is $-4$ , so we can add or subtract $11$ to find possible values for $a$ .
Calculating Possible Values for $a$ : Calculating the possible values for $a$ , we get $-4 + 11 = 7$ and $-4 - 11 = -15$ . However, $-15$ is not one of the options given, so we consider the value $7$ and check if it is within the options provided.
Re-evaluating Calculation: The value $7$ is not listed in the options, so we must have made a mistake in our previous step. We need to re-evaluate our calculation.