Prove that the straight lines joining the origin to the points of intersection of the straight line `hx+ky=2hk` and the curve `(x-k)^(2)+(y-h)^(2)=c^(2)` are at right angle if `h^(2)+k^(2)=c^(2)`.

Prove that the straight lines joining the origin to the points of inte

1.	Prove that the straight lines joining the origin to the points of intersection of the straight line `hx+ky=2hk` and the curve `(x-k)^(2)+(y-h)^(2)=c^(2)` are at right angle if `h^(2)+k^(2)=c^(2)`.
Answer» 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)`

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