Question

The locus of the mid point of the intercept of the line $x\cos \alpha +y\sin \alpha =p$ between the coordinate axes is

• A. $x^{-2}+y^{-2}=4p^{-2}$
• B. $x^{-2}+y^{-2}=p^{-2}$
• C. $x^2+y^2=4p^{-2}$
• D. $x^2+y^2=p^{-2}$

Answer 1

The correct option is A $x^{-2}+y^{-2}=4p^{-2}$

Consider the line segment $x\cos \alpha +y\sin \alpha =p$
Therefore it cuts the x axis and y axis at points $(p\sec \alpha ,0)$ and $(0,pcosec\alpha )$
Hence the coordinates of the midpoint are $(\frac{p\sec \alpha }{2},\frac{pcosec\alpha }{2})$
Therefore
$x=\frac{p\sec \alpha }{2}$ and $y=\frac{pcosec\alpha }{2}$
Or
$\cos \alpha =\frac{p}{2x}$ and $sin\alpha =\frac{p}{2y}$
Since
${\cos }^2\alpha +si{n}^2\alpha =1$
$(\frac{p}{2x}{)}^2+(\frac{p}{2y}{)}^2=1$
$x^{-2}+y^{-2}=4p^{-2}$