Question

The number of integral values of \( \lambda \) for which the equation \[ x^2 + y^2 - 2\lambda x + 2\lambda y + 14 = 0 \] represents a circle whose radius cannot exceed 6 is:

• A. 10
• B. 11
• C. 12
• D. 9

Hint:

When completing the square, ensure that both \(x\) and \(y\) terms are perfectly squared to form a standard circle equation.

Answer 1

The Correct Option is B

The given equation is: \[ x^2 + y^2 - 2\lambda x + 2\lambda y + 14 = 0 \] Rewriting this in standard form: \[ (x^2 - 2\lambda x) + (y^2 + 2\lambda y) = -14 \] Complete the square for both \(x\) and \(y\): \[ (x^2 - 2\lambda x + \lambda^2) + (y^2 + 2\lambda y + \lambda^2) = -14 + 2\lambda^2 \] \[ (x - \lambda)^2 + (y + \lambda)^2 = 2\lambda^2 - 14 \] For this equation to represent a circle, the radius must be non-negative: \[ 2\lambda^2 - 14 \geq 0 \quad \Rightarrow \quad \lambda^2 \geq 7 \] \[ |\lambda| \geq \sqrt{7} \] The radius is given by: \[ r = \sqrt{2\lambda^2 - 14} \] To ensure the radius does not exceed 6: \[ \sqrt{2\lambda^2 - 14} \leq 6 \quad \Rightarrow \quad 2\lambda^2 - 14 \leq 36 \] \[ 2\lambda^2 \leq 50 \quad \Rightarrow \quad \lambda^2 \leq 25 \] Thus: \[ \sqrt{7} \leq |\lambda| \leq 5 \] The integer values of \( \lambda \) are \( \lambda = -5, -4, -3, -2, -1, 1, 2, 3, 4, 5 \), so there are 11 integer values of \( \lambda \).