Circles
Tangent Lines — Locus of Intersection Point
nta_pyq_2026_jan
Grade 11
Question:
Let PQ and MN be two straight lines touching the circle $x^2+y^2-4x-6y-3=0$ at the points A and B respectively. Let O be the centre of the circle and $\angle AOB=\pi/3$. Then the locus of the point of intersection of the lines PQ and MN is:
x^2+y^2-12x-18y-25=0
3(x^2+y^2)-18x-12y+25=0
3(x^2+y^2)-12x-18y-25=0
x^2+y^2-18x-12y-25=0
Step-by-Step Solution
Key Concept: Circle: centre $O=(2,3)$, radius $r=4$. PQ and MN are tangents at A and B with $\angle AOB=\pi/3$. For external point P with tangent length: $\sin(\pi/3)=4/OP\Rightarrow OP=8/\sqrt{3}$.
Locus: $3(x^2+y^2)-12x-18y-25=0$.
Correct Answer: 3