Question

If Q is a point on the locus of x² + y² + 4x – 3y +7 = 0, then find the equation of locus of P which divides segment OQ externally in the ratio 3 : 4 where O is origin

Answer 1

Let Q be (a, b) lying on the locus

x² + y² + 4x – 3y + 7 = 0

∴ a² + b² + 4a – 3b + 7 = 0

Let the movable point P be (h, k)

Given P divides OQ externally in the ratio 3 : 4

(h, k) = `((3"a" - 4 xx 0)/(3 - 4), (3"b" - 4 xx 0)/(3 - 4))`

(h, k) = `((3"a")/(- 1), (3"b")/(- 1))`

h = – 3a, k = – 3b

a = `- "h"/3`, b = `- "k"/3`

Substituting in equation (1) we have

`(- "h"/3)^2 + (- "k"/3)^2 + 4(- "h"/3) - 3(- "k"3) + 7`  0

`"h"^2/9 + "k"^2/9 - (4"h")/3 + (3"k")/3 + 7` = 0

h² + k² – 12h + 9k + 63 = 0

The locus of P(h, k) is obtained by replacing h by x and k by y.

∴ The required locus is x² + y² – 12x + 9y + 63 = 0