Question

From point P on the line x + y - 25 = 0, tangents PA and PB are drawn to the circle x² + y² = 9. As point P moves on the line, the locus of the midpoint of the chord AB is

(A)  5(x² + y²) = 9(x + y)

(B)  5(x² + y²) = 3(x + y)

(C)  25(x² + y²) = 3(x + y)

(D)  25(x² + y²) = 3(x + y)

Answer 1

Correct option  (C)  25(x² + y²) = 3(x + y)

Explanation :

 Let P(h, k) be a point on the line x + y - 25 = 0. Therefore

h + k = 25   ....(1)

 Since AB being the chord of contact of P with respect to the circle x² + y² = 9, its equation is

hx + ky = 9  ....(2)

 If M(x₁, y₁) is the midpoint of the chord AB, its equation is

 xx₁ +  yy₁ =  x²₁ + y₁² ......(3)

 Both Eqs. (2) and (3) represent the chord AB. Therefore