Prove that the straight lines joining the origin to the point of intersection of the straight line `h x+k y=2h k` and the curve `(x-k)^2+(y-h)^2=c^2` are perpendicular to each other if `h^2+k^2=c^2dot`

Explanation

From the equation of line , we have
`1=(hx+ky)/(2hx)` (1)
Equation of the curve is
`x^(2)+y^(2)-2kx-2hy+h^(2)+k^(2)-c^(2)=0`
Making above equation homogeneous with the help of (1) , we get
`x^(2)+y^(2)-2(kx+hy)((hx+ky)/(2hx))+(h^(2)+k^(2)-c^(2))((hx+ky)/(2hk))^(2)=0`
This is combined equation of the pair of lines joining the origin to the points of intersection of the given line and the curve . the componenet lines are perpendicular if sum of coefficient of `x^(2)` of and coefficient of `y^(2)` is zero .
` :. (h^(2)+k^(2))(h^(2)+k^(2)-c^(2))=0`
`rArr h^(2)+k^(2)=c^(2)`
`(ash^(2)+k^(2)ne0)`