Q46 / 160 · AP EAMCET 2014 · Mathematics
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
- Correct answerA$\sqrt{41} \mathrm{~cm}$
- B$\sqrt{11} \mathrm{~cm}$
- C$\sqrt{21} \mathrm{~cm}$
- D$\sqrt{31} \mathrm{~cm}$
Solution
Let $r$ be the radius of the circle.
Now, perpendicular distance $$ \begin{aligned} A C & =\frac{|2+4+2|}{\sqrt{1^2+1^2}}=\frac{8}{\sqrt{2}} \\ & =4 \sqrt{2} \end{aligned} $$ In right angled $\triangle C A B$, $$ \begin{aligned} r^2 & =(A C)^2+(A B)^2 \\ & =(4 \sqrt{2})^2+(3)^2=32+9 \\ \Rightarrow \quad r^2 & =41 \Rightarrow r=\sqrt{41} \end{aligned} $$
Now, perpendicular distance $$ \begin{aligned} A C & =\frac{|2+4+2|}{\sqrt{1^2+1^2}}=\frac{8}{\sqrt{2}} \\ & =4 \sqrt{2} \end{aligned} $$ In right angled $\triangle C A B$, $$ \begin{aligned} r^2 & =(A C)^2+(A B)^2 \\ & =(4 \sqrt{2})^2+(3)^2=32+9 \\ \Rightarrow \quad r^2 & =41 \Rightarrow r=\sqrt{41} \end{aligned} $$
Hence, the correct answer is option: 'A'.