Question

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

Answer 1

Given equation of tangents are-

$T_1:3x-4y+4=0.....\left(i\right)$

$T_2:6x-8y-7=0$

$\Rightarrow 3x-4y-\frac{7}{2}=0.....\left(ii\right)$

The slope of of given tangents is same, i.e., $\frac{3}{4}$, thus the given tangents are parallel.

As we know that the distance between the two parallel tangents of a circle is equal to the diameter of circle.

Therefore,

$T_1{T}_2=\frac{|(-\frac{7}{2})-4|}{\sqrt{{\left(3\right)}^2+{(-4)}^2}}$

$\Rightarrow T_1{T}_2=\frac{|-\frac{15}{2}|}{\sqrt{9+16}}$

$\Rightarrow T_1{T}_2=\frac{15}{2\times 5}=\frac{3}{2}$

Therefore,

Radius of the circle $=\frac{T_1{T}_2}{2}=\frac{3}{4}\text{units}$

Hence the radius of the circle is $\frac{3}{4}$ units.