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
The area of the circle $x^2 + y^2 + 8x - 6y + c = 0$ is $75\pi$. Then the value of $c$ is equal to
Show 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.
Updated On: Jun 26, 2026
- -50
- 50
- 25
- -25
- -40
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Concept:
The general equation of a circle is \( x^2 + y^2 + 2gx + 2fy + c = 0 \). The radius is \( r = \sqrt{g^2 + f^2 - c} \). The area of the circle is \( \pi r^2 \).
Step 2: Detailed Explanation:
From the equation \( x^2 + y^2 + 8x - 6y + c = 0 \):
\( 2g = 8 \implies g = 4 \)
\( 2f = -6 \implies f = -3 \)
The radius squared is \( r^2 = g^2 + f^2 - c = 4^2 + (-3)^2 - c = 16 + 9 - c = 25 - c \).
Given Area = \( 75\pi \).
\[ \pi r^2 = 75\pi \]
\[ r^2 = 75 \]
Equating the two expressions for \( r^2 \):
\[ 25 - c = 75 \]
\[ -c = 75 - 25 \]
\[ -c = 50 \]
\[ c = -50 \]
Step 3: Final Answer:
The value of $c$ is -50.
The general equation of a circle is \( x^2 + y^2 + 2gx + 2fy + c = 0 \). The radius is \( r = \sqrt{g^2 + f^2 - c} \). The area of the circle is \( \pi r^2 \).
Step 2: Detailed Explanation:
From the equation \( x^2 + y^2 + 8x - 6y + c = 0 \):
\( 2g = 8 \implies g = 4 \)
\( 2f = -6 \implies f = -3 \)
The radius squared is \( r^2 = g^2 + f^2 - c = 4^2 + (-3)^2 - c = 16 + 9 - c = 25 - c \).
Given Area = \( 75\pi \).
\[ \pi r^2 = 75\pi \]
\[ r^2 = 75 \]
Equating the two expressions for \( r^2 \):
\[ 25 - c = 75 \]
\[ -c = 75 - 25 \]
\[ -c = 50 \]
\[ c = -50 \]
Step 3: Final Answer:
The value of $c$ is -50.
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