Question

If the line $lx+my=1$ is a tangent to the circle $x^2+y^2=a^2$, then point (l,m) lies-

• A. outside the circle
• B. on the circle
• C. far away from the circle
• D. none of these

Answer 1

The correct option is B on the circle

If the line $lx+my-1=0$ touches the circle $x^2+y^2=a^2$, then length of the perpendicular from its centre O(0,0) is equal to the radius a.

Therefore,

$|\frac{m+m\times 0-1}{\sqrt{l^2+m^2}}|=a$

$l^2+m^2=\frac{1}{a^2}$

$(l,m)$ satisfies the equation $x^2+y^2=\frac{1}{a^2}$.

$(l,m)$ lies on the circle $x^2+y^2=\frac{1}{a^2}$.

Hence, $(l,m)$ lies on a circle.