Question

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

Answer 1

Given lines are $3x-4y+4=0$ and $6x-8y-7=0.$

$\Rightarrow 3x-4y-\frac{7}{2}=0$ are tangents to the circle.

Also, given lines are parallel, because their slopes are equal, i.e.$m_1=\frac{3}{4}=m_2$

$\therefore$ Radius = $\frac{1}{2}\times$ (Distance between two parallel lines)

= $\frac{1}{2}\left|\frac{4+\frac{7}{2}}{\sqrt{3^2+4^2}}\right|$

= $\frac{1}{2}-\frac{\left|\frac{15}{2}\right|}{\sqrt{9+16}}=\frac{15/2}{2\times 5}=\frac{3}{4}$