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Tangents `P A` and `P B` are drawn to `x^2+y^2=9` from any arbitrary point `P` on the line `x+y=25` . The locus of the midpoint of chord `A B` is (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

A

`x^(2)+y^(2) -2x -2y = 0`

B

`x^(2) +y^(2) +2x +2y = 0`

C

`x^(2) +y^(2) - 2x +2y = 0`

D

`x^(2) +y^(2) +2x - 2y= 0`

Text Solution

Verified by Experts

The correct Answer is:
A
Let `P(a,4-a)`.

The equation of chord of contact AB is
`xa +y (4-a) =8` (i)
Also, equation of chord AB whose mid-points is `M(h,k)` is
`hx + ky = h^(2) +k^(2)` (ii) (using `T = S_(1))`
Since (i) and (ii) are identical, on comparing the coefficients, we get
`(a)/(h) =(4-a)/(k) = (8)/(h^(2) +k^(2)) =(a+4-a)/(h+k) =(4)/(h+k)`
`rArr 4(h^(2)+k^(2)) =8(h+k)`
Hence, locus of `M(h,k)` is `x^(2) +y^(2) -2x -2y =0`.
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