If the lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangent to the same circle. The radius of the circle is __________.Correct answer is `3/4`. Can you explain this answer?

Question

Answer

Understanding the Problem
To find the radius of a circle that is tangent to two given lines, we can use the formula for the distance from a point to a line. The lines provided are:
1. 3x - 4y + 4 = 0
2. 6x - 8y - 7 = 0
Step 1: Simplifying the Lines
- The second line can be simplified. Notice that it is a multiple of the first line. We can rewrite it as:
3x - 4y - 3.5 = 0
This indicates that both lines are parallel.
Step 2: Finding the Distance Between the Lines
- The distance (D) between two parallel lines of the form Ax + By + C1 = 0 and Ax + By + C2 = 0 is given by:
D = |C2 - C1| / sqrt(A^2 + B^2)
- For our lines:
C1 = 4, C2 = -3.5, A = 3, B = -4
- Plugging in the values, we calculate:
D = |(-3.5) - (4)| / sqrt(3^2 + (-4)^2)
= | -7.5 | / sqrt(9 + 16)
= 7.5 / 5
= 1.5
Step 3: Radius of the Circle
- Since both lines are tangents to the circle, the radius (r) of the circle is half the distance between the two tangents:
r = D / 2 = 1.5 / 2 = 0.75
- Thus, converting to fraction form:
r = 3/4
Final Answer
The radius of the circle is 3/4.

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