CAT 2023 Coordinate Geometry Question (Hard)

Question

Let be the circle and be the locus of the point of intersection of a pair of tangents to with the angle between the two tangents equal to . Then, the point at which touches the line is

Accepted Answer

Here is the stepwise solution with the image link included --- Step 1 Given circle equation x 2 y 2 4x - 6y - 3 0 We rewrite it in standard form by completing the squares. --- Step 2 Complete the square for x 2 4x x 2 2 cdot 2 cdot x 2 2 - 2 2 x 2 2 - 4 --- Step 3 Complete the square for y 2 - 6y y

CAT 2023

Quantitative

Coordinate Geometry

Hard

Let $\mathrm{C}$ be the circle $x^2+y^2+4x-6y-3=0$ and $\mathrm{L}$ be the locus of the point of intersection of a pair of tangents to $\mathrm{C}$ with the angle between the two tangents equal to $60^{\circ}$ . Then, the point at which $L$ touches the line $x=6$ is

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