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

B

Explanation

Let P(t, 4-t) be an arbitrary point on the line x+y=4 and let Q (h, k) be the mid-point of the chord of contact AB of tangents drawn from P.
The equation of AB regarded as the chord of contact of tangents drawn from P is
`tx + (4-t)y=8 " " ...(i)`
The equation of chord AB of the circle `x^(2)+y^(2)=8` when Q(h, k) is the mid-point, is
`hx+ky=h^(2)+k^(2) " " ...(ii) `
Clearly, equations (i) and (ii) represent the same line.
`:. (t)/(h)=(4-t)/(k)=(8)/(h^(2)+k^(2))`
`rArr t (h^(2)+k^(2))=8h and t=(4h)/(h+k)`
On laminating t, we get
`h^(2)+k^(2)=2(h+k)`
Hence, the locus of (h, k) is `x^(2)+y^(2)=2(x+y)`.