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The locus of the mid-point of the chord ...

The locus of the mid-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) `20(x^2+y^2)-20x+45y=0` (D) `20(x^2+y^2)+20x-45y=0`

A

`20(x^(2)+y^(2))-36+45y=0`

B

`20(x^(2)+y^(2))+36-45y=0`

C

`36(x^(2)+y^(2))-20x+45y=0`

D

`36(x^(2)+y^(2))+20x-45y=0`

Text Solution

Verified by Experts

The correct Answer is:
A

The equation of the chord AB bisected at `P(h,k)` is
`hx+ky= h^(2)+k^(2)` (1)
Let any point on the given line be
`(alpha, (4)/(5)alpha -4)`
The equation of the chord of contact AB is
`alpha x + ((4)/(5) alpha -4) y =9` (2)
Comparing (1) and (2) , we get
`(h)/(alpha)=(k)/((4//5)alpha-4)=(h^(2)+k^(2))/(9)`
`:. alpha =(20h)/(4h-5k)` ( From `1^(st)` and `2^(nd)` ratio)
`:. (h(4h-5k))/(20h)=(h^(2)+k^(2))/(9)`
or `20(h^(2)+k^(2))=9(4h-5k)`
or `20(x^(2)+y^(2))-36x+45y=0`
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