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:
(A) x² + y² – 16x²y² = 0
(B) x² + y² – 2x²y² = 0
(C) x² + y² – 4x²y² = 0
(D) x² + y² – 2xy = 0

Answer 1

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 given by,

Let us consider the standard equation of tangent to be,

x cos ∅ + y sin ∅ = 1

Therefore, the coordinates of P and Q are given by,

$P(\frac{1}{cos \theta} , 0)$

$Q(0,\frac{1}{sin \theta})$

Let the mid-point of PQ has the coordinates (h, k)

Therefore, we get,

$h = \dfrac{\frac{1}{cos \theta} + 0 }{2}$  and

$k = \dfrac{0 + \frac{1}{sin \theta} }{2}$

⇒ cos ∅ = 1/2h

sin ∅ = 1/2k

squaring and adding equations (1) and (2), we get,

cos² ∅ + sin² ∅ = 1/4h² + 1/4k²

1 =  1/4h² + 1/4k²

Therefore, the equation of locus is given by,

⇒ 1/4x² + 1/4y² = 1

⇒ 4x² + 4y² = 4xy

∴ x² + y² - 4xy = 0