Question

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

Answer 1

We have 3x − 4y + 4 = 0 and 6x − 8y − 7 = 0

\[\Rightarrow y = \frac{3}{4}x + 1 and y = \frac{3}{4}x - \frac{7}{8}\]

Since, the slope of both the lines are equal.
Hence, the both the lines are parallel.
The distance between the parralel lines is given by

\[\left| \frac{C_1 - C_2}{\sqrt{A^2 + B^2}} \right|\]
\[ = \left| \frac{4 + \frac{7}{2}}{\sqrt{3^2 + 4^2}} \right|\]
\[ = \left| \frac{\frac{15}{2}}{5} \right|\]
\[ = \frac{3}{2}\]

Now, the radius is equal to the half of the distance between the parallel lines(diameter of the circle).
Hence, the radius is given by

\[\frac{3}{4}\]