Question

The area of the circle \( x^2 - 2x + y^2 - 10y + k = 0 \) is \( 25\pi \). The value of \( k \) is equal to:

• A. \( -1 \)
• B. \( 1 \)
• C. \( 0 \)
• D. \( 2 \)
• E. \( 3 \)

Hint:

Completing the square is an alternative: \( (x^2 - 2x + 1) + (y^2 - 10y + 25) = 1 + 25 - k \). This gives \( (x-1)^2 + (y-5)^2 = 26 - k \). Since \( r^2 = 25 \), we have \( 26 - k = 25 \), so \( k = 1 \).

Answer 1

The Correct Option is B

Concept: The area of a circle is given by \( A = \pi r^2 \). By comparing the given area to this formula, we find the radius \( r \). We then use the general equation of a circle \( x^2 + y^2 + 2gx + 2fy + c = 0 \), where the radius is \( r = \sqrt{g^2 + f^2 - c} \), to solve for the unknown constant \( k \).

Step 1: Determine the radius from the given area.
Given Area \( = 25\pi \). \[ \pi r^2 = 25\pi \implies r^2 = 25 \implies r = 5 \]

Step 2: Identify the coefficients from the circle's equation.
The given equation is \( x^2 + y^2 - 2x - 10y + k = 0 \). Comparing with \( x^2 + y^2 + 2gx + 2fy + c = 0 \):
• \( 2g = -2 \implies g = -1 \)
• \( 2f = -10 \implies f = -5 \)
• \( c = k \)

Step 3: Solve for \( k \) using the radius formula.
We know \( r^2 = g^2 + f^2 - c \): \[ 25 = (-1)^2 + (-5)^2 - k \] \[ 25 = 1 + 25 - k \] \[ 25 = 26 - k \] \[ k = 26 - 25 = 1 \]