Question

What is the condition for the line y = mx + c to be a secant of the circle $x^2+y^2=a^2$

• A. $a^2(1+m^2)<c^2$
• B. $a^2(1+m^2)>c^2$
• C. $a^2(1+m^2)=c^2$
• D. $c<a^2(1+m^2)$

Answer 1

The correct option is B

$a^2(1+m^2)>c^2$

We will first draw the circle and line to figure out the condition

If L is a secant, the perpendicular distance of centre c from L will be less than the radius.

$\Rightarrow cp<r=a$ .........(1)

We know that the centre of the circle $x^2+y^2=a^2$ is (0,0).

cp=distance of (0,0) from $y=mx+c$

=$\left|\frac{c}{\sqrt{1+m^2}}\right|$

From (1),$\left|\frac{c}{\sqrt{1+m^2}}\right|<a$

$\Rightarrow c^2<a^2(1+m^2)$