Question

If \[ S = 2x^2 + 2y^2 - 8x + 8y - 7 = 0 \] is the circle passing through the points of intersection of the circles \[ x^2 + y^2 - kx - ky + 1 = 0 \quad \text{and} \quad x^2 + y^2 - kx + ky - 2 = 0, \] then the length of the tangent drawn from the point \( (k, k) \) to the circle \( S \) is:

• A. \( \frac{3}{\sqrt{2}} \)
• B. \( 3 \)
• C. \( \frac{\sqrt{23}}{2} \)
• D. \( \sqrt{23} \)

Hint:

Use formula for tangent length: \( \sqrt{(x - a)^2 + (y - b)^2 - r^2} \) or geometric radius when point lies symmetrically.

Answer 1

The Correct Option is A

Standard method: Equation of circle is: \[ 2x^2 + 2y^2 - 8x + 8y - 7 = 0 \Rightarrow x^2 + y^2 - 4x + 4y = \frac{7}{2} \] Center = \( (2, -2) \), radius = \( \sqrt{2^2 + (-2)^2 - \frac{7}{2}} = \sqrt{8 - \frac{7}{2}} = \sqrt{\frac{9}{2}} = \frac{3}{\sqrt{2}} \) Distance from point \( (k, k) \) to this circle = length of tangent = radius (since point lies on radical axis) \[ \boxed{\text{Tangent length} = \frac{3}{\sqrt{2}}} \]