Question

The point of intersection of the tangents drawn at the points where the line \[ 2x - y + 3 = 0 \] meets the circle \[ x^2 + y^2 - 4x - 6y + 4 = 0 \] is:

• A. \( \left(-8, \frac{15}{2}\right) \)
• B. \( \left(-\frac{5}{2}, \frac{21}{4}\right) \)
• C. \( \left(\frac{5}{2}, -\frac{21}{4}\right) \)
• D. \( \left(8, -\frac{15}{2} \right) \)

Hint:

Intersection of tangents at points on a circle from a secant line lies on the polar line — use coordinate geometry tools to find it.

Answer 1

The Correct Option is B

Given: - Line intersects circle at two points ⇒ tangents drawn at those points. - Required point is intersection of those tangents ⇒ this is known as the polar of the point. Alternative approach: - Find points of intersection between line and circle - Find equations of tangents at these points - Solve system to get the point of intersection After full algebraic elimination: \[ \boxed{\left(-\frac{5}{2}, \frac{21}{4}\right)} \]