Question

The area of the circle $x^2 + y^2 + 8x - 6y + c = 0$ is $75\pi$. Then the value of $c$ is equal to

• A. -50
• B. 50
• C. 25
• D. -25
• E. -40

Hint:

Remember that Area/\( \pi \) gives you \( r^2 \). Then just use the radius formula \( r^2 = g^2 + f^2 - c \) to solve for the missing constant.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The general equation of a circle is \(x^2 + y^2 + 2gx + 2fy + c = 0\), with radius \(r = \sqrt{g^2 + f^2 - c}\).
We relate the given area to the formula for the radius.
Step 2: Key Formula or Approach:
The area of a circle is \(\pi r^2\).
Set \(\pi r^2 = 75\pi\) to find \(r^2\).
Then use \(r^2 = g^2 + f^2 - c\) to solve for \(c\).
Step 3: Detailed Explanation:
From the area:
\[ \pi r^2 = 75\pi \implies r^2 = 75 \] From the circle's equation \(x^2 + y^2 + 8x - 6y + c = 0\):
\(2g = 8 \implies g = 4\)
\(2f = -6 \implies f = -3\)
The radius squared is:
\[ r^2 = g^2 + f^2 - c \] Substitute the known values:
\[ 75 = 4^2 + (-3)^2 - c \] \[ 75 = 16 + 9 - c \] \[ 75 = 25 - c \] Rearrange to solve for \(c\):
\[ c = 25 - 75 = -50 \] Step 4: Final Answer:
The value of \(c\) is -50.