Tangents `PA and PB` are drawn to the circle `x^2 +y^2=8` from any arbitrary point P on the line `x+y =4`. The locus of mid-point of chord of contact AB is

Options

A. `x^(2)+y^(2) -2x -2y = 0`
B. `x^(2) +y^(2) +2x +2y = 0`
C. `x^(2) +y^(2) - 2x +2y = 0`
D. `x^(2) +y^(2) +2x - 2y= 0`

Answer

A

Explanation

Let `P(a,4-a)`.

The equation of chord of contact AB is
`xa +y (4-a) =8` (i)
Also, equation of chord AB whose mid-points is `M(h,k)` is
`hx + ky = h^(2) +k^(2)` (ii) (using `T = S_(1))`
Since (i) and (ii) are identical, on comparing the coefficients, we get
`(a)/(h) =(4-a)/(k) = (8)/(h^(2) +k^(2)) =(a+4-a)/(h+k) =(4)/(h+k)`
`rArr 4(h^(2)+k^(2)) =8(h+k)`
Hence, locus of `M(h,k)` is `x^(2) +y^(2) -2x -2y =0`.