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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
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Verified Answer
The correct answer is:
$\sqrt{41} \mathrm{~cm}$
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 $\triangle C A B$,
$$
\begin{aligned}
r^2 & =(A C)^2+(A B)^2 \\
& =(4 \sqrt{2})^2+(3)^2=32+9 \\
\Rightarrow \quad r^2 & =41 \Rightarrow r=\sqrt{41}
\end{aligned}
$$
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