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`

Options

A. `20(x^(2)+y6(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`

Answer

A

Explanation

Let P `(t, (4t-20)/(5))` be a point on the line `4x-5y=20`. Then, the chord of contact of tangents drawn from P to the circle `x^(2)+y^(2)=9` is
`tx+((4t-20)/(5))y=9 " " ...(i)`
Let Q(h, k) be the mid-point of this chord of contact. Them, its equation is also
`hx+ky=h^(2)+k^(2)` [Using T=S'] ... (ii)
Clearly, (i) and (ii) represent the same line.
`:. (t)/(h)=(4t-20)/(5k)=(9)/(h^(2)+k^(2))`
`rArr (t)/(h) = (9)/(h^(2)+k^(2)) and (t)/(h) = (4t-20)/(5k)`
`rArr t=(9h)/(h^(2)+k^(2)) and t=(20h)/(4h-5k)`
`rArr (9h)/(h^(2)+k^(2))=(20h)/(4h-5k)`
`rArr h { 20 (h^(2)+k^(2))-36h+45k}=0`
`rArr h{20(h^(2)+k^(2))-36 h +45k}=0`
Hence, the locus of (h, k) is
`x{20(x^(2)+y^(2))-36 x + 45 y} = 0`
`rArr x=0 or, 20 (x^(2)+y^(2))-36x+45y=0`