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. `25(x^(2)+y^(2))=9(x+y)`
B. `25(x^(2)+y^(2))=3(x+y)`
C. `5(x^(2)+y^(2))=3(x+y)`
D. none of these

Answer

1

Explanation

Any point on the line `x+y=25` is `P -= (a,25-a), a in R`.
The equation of chord AB is
`T=0`
i.e.,` xa+y(25-a)=9` (1)
If the midpoint of chord AB is C(h,k), then the equation of chord AB is
`T=S_(1)`
i.e., `xh+yk=h^(2)k^(2)` (2)
Comparing the ratio of coefficients of (1) and (2), we get
`(a)/(h)=(25-a)/(k)=(9)/(h^(2) +k^(2))`
or `(a+25-a)/(h+k)=(9)/(h^(2)+k^(2))`
Thus,t he locus of C is `25(x^(2)=y^(2))=9(x+y)`