9.Straight Line
medium

બિંદુઓ $(1, 0)$ અને $(2\cos \theta ,2\sin \theta )$ ને જોડતા રેખાખંડને $2 : 3$ ગુણોતરમાં અંત:વિભાજન કરતા બિંદુ $P$ ના બિંદુપથનું સમીકરણ  . . . . દર્શાવે.

A

રેખા

B

વર્તૂળ

C

રેખાઓ

D

પરવલય

(IIT-1986)

Solution

(b) Let the coordinates of the point $P$ which divides the line joining $(1, 0)$ and $(2\cos \theta ,\,2\sin \theta )$in the ratio $2:3$ be $(h,k)$. Then, $h = \frac{{4\cos \theta + 3}}{5}$and $k = \frac{{4\sin \theta }}{5}$

==> $\cos \theta = \frac{{5h – 3}}{4}$and $\sin \theta = \frac{{5k}}{4}$

==>${\left( {\frac{{5h – 3}}{4}} \right)^2} + {\left( {\frac{{5k}}{4}} \right)^2} = 1$==>${(5h – 3)^2} + (5{k^2}) = 16$

Therefore locus of $(h,k)$ is ${(5x – 3)^2} + {(5y)^2} = 16$,which is $a$ circle.

Standard 11
Mathematics

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