CAT 2023 Slot 1 QA Q17 Coordinate Geometry

Question 17 of the Quantitative Ability (QA) section, CAT 2023 Slot 1, with CAT's official answer and a worked solution.

Let \(\mathrm{C}\) be the circle \(x^2+y^2+4 x-6 y-3=0\) 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

A
(6,4)
B
(6,8)
C
(6,3)
D
(6,6)

Answer: (C) (6,3) Coordinate Geometry MCQ

Solution

Rewrite: \((x+2)^2 + (y-3)^2 = 16\); centre \((-2,3)\), radius 4.
If tangents from P meet at \(60^\circ\), the line to the centre bisects it: \(\sin30^\circ = \frac{4}{d}\), so \(d = 8\).
Locus L is the circle with centre \((-2,3)\) and radius 8.
The line \(x=6\) is at distance \(6-(-2)=8\) from the centre, so it touches L at the foot of the perpendicular, \((6,3)\).

Shortcut. The touching point must have the centre's \(y\)-coordinate, 3.