Coordinate Geometry Question 104
Question: The locus of the mid-point of the distance between the axes of the variable line $ x\cos \alpha +y\sin \alpha =p, $ where p is constant, is
[MNR 1985; CEE 1993; UPSEAT 2000; AIEEE 2002]
Options:
A) $ x^{2}+y^{2}=4p^{2} $
B) $ \frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{4}{p^{2}} $
C) $ x^{2}+y^{2}=\frac{4}{p^{2}} $
D) $ \frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{2}{p^{2}} $
Show Answer
Answer:
Correct Answer: B
Solution:
The straight line $ x\cos \alpha +y\sin \alpha =p $ meets the x-axis at the point $ A( \frac{p}{\cos \alpha },0 ) $ and the y-axis at the point $ B( 0,\frac{p}{\sin \alpha } ) $ . Let (h, k) be the coordinates of the middle point of the line segment AB. Then, $ h=\frac{p}{2\cos \alpha } $ and $ k=\frac{p}{2\sin \alpha } $
$ \Rightarrow \cos \alpha =\frac{p}{2h} $ and $ \sin \alpha =\frac{p}{2k} $
$ \Rightarrow {{\sin }^{2}}\alpha +{{\cos }^{2}}\alpha =\frac{p^{2}}{4h^{2}}+\frac{p^{2}}{4k^{2}}=1 $
Hence locus of the point (h, k) is $ \frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{4}{p^{2}} $ .
BETA
