Multiple choice

If the lines $3x-4y+4=0$ and $6x-8y-7=0$ are tangents to a circle, then the diameter of the circle

  1. $\dfrac{3}{2}$
  2. $3$
  3. $\dfrac{5}{2}$
  4. $5$
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A Correct answer
Explanation

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 is |c1 - c2| / sqrt(a^2 + b^2) = |4 - (-3.5)| / sqrt(3^2 + (-4)^2) = 7.5 / 5 = 1.5. The diameter is 1.5 = 3/2.

AI explanation

The equations of the lines are 3x - 4y + 4 = 0 and 3x - 4y - 7/2 = 0, which are parallel. The distance between these two parallel tangent lines gives the diameter of the circle. Using the distance formula d = |c2 - c1| / sqrt(a^2 + b^2), we get d = |-7/2 - 4| / sqrt(3^2 + (-4)^2) = 15/2 / 5 = 3/2.

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