Question

If the straight line `x - 2y + 1 = 0` intersects the circle `x^2 + y^2 = 25` at points P and Q, then find the coordinates of the point of intersection of the tangents drawn at P and Q to the circle `x^2 + y^2 = 25`.

Answer 1

Let R(h,k) be the point of intersection of the tangents drawn at P and Q to the given circle. Then PQ is the chord of the cord of tangents drawn from R to `x^(2)+y^(2)=25`
So, its equation is
`hx+ky-25=0` (1)
It is given that the eqation of PQ is
`x-2y+1=0` (2)
Since (1) and (2) represent the same line, we have
`(h)/(1)=(k)/(-2)=(-25)/(1)`
or `h=-25,k=50 `
Hence, the required point is `(-25,50)`