If the lines $3x-4y+4=0$ and $6x-8y-7=0$ are tangents to a circle, then find the radius of the circle.
- $3/4$
- $4/3$
- $1/4$
- $7/4$
The lines are 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0. Rewrite the second as 3x - 4y - 3.5 = 0. The distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is |c1 - c2| / sqrt(a^2 + b^2). Distance = |4 - (-3.5)| / sqrt(3^2 + 4^2) = 7.5 / 5 = 1.5. The diameter is 1.5, so the radius is 0.75 or 3/4.
First, simplify the second tangent equation by dividing it by 2, resulting in 3x - 4y - 3.5 = 0. The distance between the two parallel lines 3x - 4y + 4 = 0 and 3x - 4y - 3.5 = 0 is found using the parallel line distance formula: the absolute difference of the constants divided by the square root of (3 squared plus 4 squared). This distance evaluates to 7.5 divided by 5, which is 1.5. Since these lines are parallel tangents to the same circle, this distance equals the diameter, making the radius half of 1.5. The radius is 3/4.