Question

For what value of \( k \), does the equation \[ 9x^2 + y^2 = k(x^2 - y^2 - 2x) \] represent the equation of a circle?

• A. 1
• B. 2
• C. -1
• D. 4

Hint:

For the equation to represent a circle, the coefficients of \( x^2 \) and \( y^2 \) must be equal, and there should be no linear term in \( x \).

Answer 1

The Correct Option is D

Step 1: Rearrange the given equation to the standard form of a circle: \[ 9x^2 + y^2 = k(x^2 - y^2 - 2x). \] Expand the right-hand side: \[ 9x^2 + y^2 = kx^2 - ky^2 - 2kx. \] Move all terms involving \( x \) and \( y \) to one side: \[ 9x^2 - kx^2 + y^2 + ky^2 + 2kx = 0. \] Simplify: \[ (9 - k)x^2 + (1 + k)y^2 + 2kx = 0. \] Step 2: For the equation to represent a circle, the coefficients of \( x^2 \) and \( y^2 \) must be equal, and there should be no linear term in \( x \). Therefore, we set: \[ 9 - k = 1 + k \quad \Rightarrow \quad 2k = 8 \quad \Rightarrow \quad k = 4. \]