Question

Locus of the point of intersection of the lines `xcosalpha+ysinalpha=a` and `xsinalpha-ycosalpha=b` where `alpha` is variable.

Answer 1

Correct Answer - `x^2+y^2=a^2+b^2`
Let (h,k) be the point of intersection of `x cosalpha+ysinalpha=a` and `xsinalpha-ycosalpha=b`. Then,
`hcosalpha+ksinalpha=a` (1)
`hsinalpha-kcosalpha=b` (2)
Squareing and adding (1) and (2),we get
`(hcosalpha+ksinalpha)^2+(hsinalpha-kcosalpha)^2=a^2+b^2`
or `h^2+k^2=a^2+b^2`
Hence, the locus of (h,k) is
`x^2+k^2=a^2+b^2`