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The locus of the middle point of the cho...

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

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`

Text Solution

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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`
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