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If the lines $3 x+4 y-14=0$ and $6 x+8 y+7=0$ are both tangents to a circle, then its radius is
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Verified Answer
The correct answer is:
$\frac{7}{4}$
Given, lines
are both tangents to a circle we observe that, both lines are parallel i.e.,
$\begin{aligned}
3 x+4 y-14 & =0 \\
3 x+4 y+\frac{7}{2} & =0
\end{aligned}$
So, radius of circle $=\frac{1}{2}$ (distance between two parallel lines)
$\begin{aligned} & =\frac{1}{2}\left\{\frac{|7 / 2+14|}{\sqrt{9+16}}\right\} \\ & =\frac{1}{2}\left\{\frac{35 / 2}{5}\right\}=\frac{1}{2} \times \frac{7}{2} \\ & =\frac{7}{4}\end{aligned}$
are both tangents to a circle we observe that, both lines are parallel i.e.,
$\begin{aligned}
3 x+4 y-14 & =0 \\
3 x+4 y+\frac{7}{2} & =0
\end{aligned}$
So, radius of circle $=\frac{1}{2}$ (distance between two parallel lines)
$\begin{aligned} & =\frac{1}{2}\left\{\frac{|7 / 2+14|}{\sqrt{9+16}}\right\} \\ & =\frac{1}{2}\left\{\frac{35 / 2}{5}\right\}=\frac{1}{2} \times \frac{7}{2} \\ & =\frac{7}{4}\end{aligned}$
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