Find the locus of the middle point of the portion of the line `xcos alpha+ysin alpha=p` which is intercepted between the axes, given that `p` remains constant.

Answer

`(1)/(x^2)+(1)/(y^2)=(4)/(p^2)`

Explanation

The equation of the variable line is `xcosalpha+ysinalpha=p`
Here P is a constant and `alpha` is the parameter (variable),
Let the line in (1) cuts- x and y-axes at A and B respectively,
Putting `y=0` in (1) , we get `A-=(psecalpha,0)`
Putting `x=0` in (1) we get `B-=(0,pcosec alpha)`.
AB is the portion of (1) intercepted between the axes.
Let P(h,k) be the midpoint of AB. we have to find the locus of pont (h,k) for this, we will to eliminate `alpha` and find a relation in h and k. Therefore.
`h=(psecalpha+0)/(2)=(p)/(2)secalpha" "(2)`
and `k=(0+pcosecalpha)/(2)=(p)/(2)cosec alpha`
From (2) , we get
`cosalpha=(p)/(2h)`
From (3)we get
`sinalpha=(p)/(2k)`
or `cos^2alpha+sin^2alpha=(p^2)/(4h^2)+(p^2)/(4k^2)`
or `(1)/(h^2)+(1)/(k^2)=(4)/(p^2)`
Hence,the locus of pont P(h,k) is `(1)/(x^2)+(1)/(y^2)=(4)/(p^2)`.