Let PQ and MN be two straight lines touching the circle x^2+y^2-4 x-6 y-3=0 at the points A and B respectively. Let O be the centre of the circle and…
JEE Main 2026 — Mathematics Coordinate Geometry
2026mcqmedium
Let PQ and MN be two straight lines touching the circle x2+y2−4x−6y−3=0 at the points A and B respectively. Let O be the centre of the circle and ∠AOB=π/3. Then the locus of the point of intersection of the lines PQ and MN is :
Official previous-year question
Held on 21 Jan 2026 · Verified 6 Jul 2026.
Options
A
x2+y2−12x−18y−25=0
B
3(x2+y2)−18x−12y+25=0
C
3(x2+y2)−12x−18y−25=0
D
x2+y2−18x−12y−25=0
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Solution
Circle: (x−2)2+(y−3)2=16, center O = (2, 3), radius r=4.
For tangents from external point P touching at A and B with ∠AOB=π/3:
In quadrilateral OAPB: ∠APB=π−π/3=2π/3, so ∠OPA=π/3.
In right triangle OAP: sin(π/3)=OP4.
OP=38=383.
Locus: (x−2)2+(y−3)2=364.
3(x2+y2)−12x−18y−25=0.
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