If the lines $3x - 4y + 4 = 0$ and $6x - 8y - 7 = 0$ are tangents to a circle, then the radius of the circle is
Show Hint
Always ensure that the $x$ and $y$ coefficients are perfectly matched before applying the parallel lines distance shortcut! Once matched, computing $\frac{\Delta C}{\sqrt{A^2+B^2}}$ gives the diameter cleanly, and halving that result provides the radius.
Step 1: Notice the lines are parallel.
The lines are $3x-4y+4=0$ and $6x-8y-7=0$. Doubling the first gives $6x-8y+8=0$, which has the same $x,y$ part as the second. So they are parallel.
Step 2: Link the gap to the circle.
Two parallel tangents touch the circle on opposite sides, so the distance between them equals the diameter.
Step 3: Find the distance.
Using $6x-8y+8=0$ and $6x-8y-7=0$, \[ d=\frac{|8-(-7)|}{\sqrt{36+64}}=\frac{15}{10}=\frac{3}{2} \] So the diameter is $\frac{3}{2}$.
Step 4: Halve for the radius.
$r=\frac{1}{2}\cdot\frac{3}{2}=\frac{3}{4}$. \[ \boxed{\frac{3}{4}\text{ units}} \]