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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
Solution
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
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