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VITEEE 2010 Paper

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Question : 98 of 120

Marks: +1, -0

The line joining (5,0) to (10 cosθ , 10 sinθ) is divided internally in the ratio 2:3 at P. If θ varies, then the locus of P is

a straight line
a pair of straight lines
a circle
None of the above

Solution:

Let coordinates of P be (h,k) then

\frac{2(10\cos\theta)+3(5)}{2+3}=4\cos\theta+3

and

\frac{2(10\sin\theta)+3(0)}{2+3}=4\sin\theta

[Using the internal section formula]

⇒

\frac{h-3}{4}=\cos\theta \text{ and } \frac{k}{4}=\sin\theta

Squaring and adding both of these equations,

\frac{(h-3)^2}{16}+\frac{k^2}{16}=\cos^2\theta+\sin^2\theta

⇒

(h-3)^2+k^2=16

Therefore, locus of point P is

(x-3)^2+y^2=16

which is a circle.

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