Question

Tangents PA and PB drawn to $x^2+y^2=9$ from any 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 point on the line $x+y=25$ be $P(a,b)$
Thus equation of chord of contact AB from point P to the circle is given by,
$T=0\Rightarrow ax+by=9$ (i)
Let mid point of AB be $R(h,k)$.
Now equation of chord AB with mid point R is given by,
$T=S_1\Rightarrow hx+ky=h^2+k^2$ (ii)
Both line (i) and (ii) represents the same line AB
$\therefore \frac{a}{h}=\frac{b}{k}=\frac{9}{h^2+k^2}$
$\Rightarrow a=\frac{9h}{h^2+k^2},b=\frac{9k}{h^2+k^2}$
Also point $(a,b)$ lie on the line $x+y=25$
$\Rightarrow a+b=25\Rightarrow 25(h^2+k^2)=9(h+k)$
Hence required locus of $R(h,k)$ is given by, $25(x^2+y^2)=9(x+y)$