Question

A circle with centre at $(2,4)$ is such that the line $x+y+2=0$ cuts a chord of length $6$ . The radius of the circle is

• A. $\sqrt{41} \,cm$
• B. $\sqrt{11} \,cm$
• C. $\sqrt{21} \,cm$
• D. $\sqrt{31} \,cm$

Answer 1

Answer: (A)

Let $r$ be the radius of the circle.

Now, perpendicular distance
$A C =\frac{|2+4+2|}{\sqrt{1^{2}+1^{2}}}=\frac{8}{\sqrt{2}} $
$=4 \sqrt{2}$
In right angled $\Delta C A B$,
$r^{2} =(A C)^{2}+(A B)^{2} $
$=(4 \sqrt{2})^{2}+(3)^{2}=32+9$
$\Rightarrow r^{2} =41 $
$\Rightarrow r=\sqrt{41}$