# If the lines $3x – 4y + 4 = 0$ and $6x – 8y – 7 = 0$ are tangents to a circle, then find the radius of the circle

$\begin{array}{1 1}\large\frac{3}{2}\text{units}\\\large\frac{3}{4}\text{units}\\\large\frac{3}{8}\text{units}\\\large\frac{3}{5}\text{units}\end{array}$

Toolbox:
• If the slopes of two lines are equal,then the lines are parallel.
• The distance between the parallel tangents gives the diameter of the circle.
• Distance between the parallel lines is $d= \bigg|\large\frac{C_1-C_2}{\sqrt{A^2+B^2}}\bigg|$
Answer : $\large\frac{3}{4}$ units
The equation of the given tangents are $3x-4y+4 = 0$ and $6x-8y-7=0$
The slopes of the given tangents are $\large\frac{3}{4}$
Hence they are parallel tangents.
Here $C_1=4$ and $C_2 =-\large\frac{7}{2}$
$\therefore$ The diameter of the circle is $d=\bigg|\large\frac{4-(-\large\frac{7}{2})}{\sqrt{3^2+4^2}}\bigg|$
$\Rightarrow\bigg|\large\frac{\Large\frac{15}{2}}{5}\bigg|=\bigg|\large\frac{15}{10}\bigg|=\frac{3}{2}$
The diameter is $\large\frac{3}{2}$ units
$\therefore$ The radius is $\large\frac{3}{4}$ units.