Question

If tangents $PA$ and $PB$ are drawn to $x^2+y^2=9$ from any arbitrary point $P$ on the line $x+y=25,$ then the locus of mid point 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. $5(x^2+y^2)=9(x+y)$

Answer 1

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

Let $P\equiv (a,25-a)$.
Equation of chord $AB$ is
$T=0\Rightarrow xa+y(25-a)=9\cdots \left(1\right)$

Let the mid point of chord $AB$ be $C(h,k).$
Then equation of chord $AB$ is
$T=S_1\Rightarrow xh+yk=h^2+k^2\cdots \left(2\right)$

Comparing equation $\left(1\right)$ and $\left(2\right)$, we get
$\frac{a}{h}=\frac{25-a}{k}=\frac{9}{h^2+k^2}$
Now, $\frac{a}{h}=\frac{9}{h^2+k^2}$
$\Rightarrow a=\frac{9h}{h^2+k^2}\cdots \left(3\right)$
Also, $\frac{25-a}{k}=\frac{9}{h^2+k^2}$
$\Rightarrow 25-a=\frac{9k}{h^2+k^2}$
Using equation $\left(3\right)$, we get
$25-\frac{9h}{h^2+k^2}=\frac{9k}{h^2+k^2}\Rightarrow 9(h+k)=25(h^2+k^2)$

Hence, the required locus is
$25(x^2+y^2)=9(x+y)$