Question

Tangents PA and PB drawn to $x^2+y^2=9$ from an arbitrary point P on the line $x+y=25$ Locus of midpoint of chord AB 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

Answer 1

The correct option is A $25(x^2+y^2)=9(x+y)$

Let the coordinates of point P be $(a,25-a)$
Equation of chord AB is $T=0$
$\Rightarrow xa+y(25-a)=9$ ...(1)
Let mid-point of chord AB be C(h,k)
Then , equation of chord AB is $T=S_1$
$\Rightarrow xh+yk=h^2+k^2$ ...(2)
Comparing the coefficients , we get
$\frac{a}{h}=\frac{25-a}{k}=\frac{9}{h^2+k^2}$
$\Rightarrow a=\frac{9h}{h^2+k^2}$
$\Rightarrow 25-\frac{9h}{h^2+k^2}=\frac{9k}{h^2+k^2}$
$\Rightarrow 25(h^2+k^2)=9(h+k)$
So, locus of C is
$25(x^2+y^2)=9(x+y)$