Question

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. $36(x^2+y^2)-20x+45y=0$
• D. $36(x^2+y^2)+20x-45y=0$

Answer 1

The correct option is A $20(x^2+y^2)-36x+45y=0$

Consider a line $\alpha x+\beta y=9$ ……. (1) ( to be a circle)

But equation of circle is given by

$x^2+y^2=9......\left(2\right)$

Let the midpoint $(h,k)$ at the circle $S$.

Then by equation (1) and (2) to,

$xh+yk=h^2+k^2=9$ …….(3)

Comparing equation (1) and (3) to, we get

$\alpha =\frac{9h}{h^2+k^2}and\beta =\frac{9h}{h^2+k^2}$

Sine $\alpha and\beta$ lies on the given line of Line

$4x-5y=20$

Then,

$4\alpha -5\beta =20$ ……(4)

Put the value of $\alpha and\beta$ in equation (4),

$\frac{4\times 9h}{h^2+k^2}-\frac{5\times 9h}{h^2+k^2}=20$

$36h-45k=20(h^2+k^2)$

$20(h^2+k^2)-36h+45k=0$

The locus is a circle

$20(x^2+y^2)-36x+45y=0$

Option (A) is correct.