The locus of the middle point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle `x^2 + y^2 = 9` is

Options

A. `20(x^(2)+y^(2))-36x+45y=0`
B. `20(x^(2)+y^(2))+36x-45y=0`
C. `36(x^(2)+y^(2))-20x+45y=0`
D. `36(x^(2)+y^(2))+20x-45y=0`

Explanation

PLAN If `S:ax^(2)+2hxy+by^(2)+2gx+2fy+c`
then equation of chord bisected at `P(x_(1),y_(1)) " is "T=S_(1)`
or `"axx"_(1)+h(xy_(1)+yx_(1))+"b " yy_(1)+g(x+x_(1))+f(y+y_(1))+C`
`=ax_(1)^(2)+2hx_(1)y_(1)+by_(1)^(2)+2gx_(1)+2fy_(1)+C`
Description of Situation As equation of chord of contact is `T=0`

Here, equation of chord of contact w.r.t. P is
`xlambda+y*((4lambda-20))/(5)=9`
`5lambdax+(4lambda-20)y=45" "...(i)`

and equation of chord bisected at the point `Q(h,k)` is
`xh+yk-9=h^(2)+k^(2)-9`
`rArr xh+ky=h^(2)+k^(2)" "...(ii)`
From Eqs. (i) and (ii) we get
`(5lambda)/(h)=(4lambda-20)/(k)=(45)/(h^(2)+k^(2))`
`therefore lambda=(20h)/(4h-5k)and lambda=(9h)/(h^(2)+k^(2))`
`rArr (20h)/(4h-5k)=(9h)/(h^(2)+k^(2))`
or `20(h^(2)+k^(2))=9(4h-5k)`
or `20(x^(2)+y^(2))=36x-45y`