Question

If the straight lines $x-4y+7=0$ and $3x-12y+11=0$ are tangents to a circle. Then, the radius of the circle is

• A. $\frac{10}{3\sqrt{17}}$
• B. $\frac{5}{3\sqrt{7}}$
• C. $\frac{15}{\sqrt{17}}$
• D. $\frac{5}{3\sqrt{17}}$
• E. $\frac{5}{3\sqrt{13}}$

Answer 1

The correct option is D $\frac{5}{3\sqrt{17}}$

Given straight lines are
$x-4y+7=0$ .... (i)
and $3x-12y+11=0$
$\Rightarrow x-4y+\frac{11}{3}=0$ .... (ii)
Since, both lines are parallel to each other.
So perpendicular distance between both lines is a diameter of circle.
$\therefore \text{Diameter}=$ Distance between parallel tangents
$\Rightarrow \text{Diameter}=\frac{|7-\frac{11}{3}|}{\sqrt{1+16}}=\frac{10}{3\times \sqrt{17}}$
Therefore, required radius $=\frac{\text{Diameter}}{2}=\frac{5}{3\sqrt{17}}$.