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Question:

If a tangent to the circle x² + y² = 1 intersects the coordinate axes at distinct points P and Q, then the locus of the mid-point of PQ is?

x²+y²−4x²y²=0

x²+y²−2xy=0

x²+y²−2xy=0

x²+y²−16x²y²=0

Solution:

The correct option is x²+y²−4x²y²=0
Let the mid point be S(h,k) ∴P(2h,0) and Q(0,2k)
equation of PQ: x/2h + y/2k = 1 ≡ PQ is tangent to circle at R(say)
∴OR=1 ⇒ √(1/2h)² + (1/2k)² = 1 ⇒ 1/4h² + 1/4k² = 1 ⇒ x² + y² − 4x²y² = 0
Aliter: tangent to circle xcosθ + ysinθ = 1
P: (secθ, 0)
Q: (0, cosecθ)
2h = secθ ⇒ cosθ = 1/2h
2k = cosecθ ⇒ sinθ = 1/2k
1/(2x)² + 1/(2y)² = 1