Multiple choice

The lines 3x - 4y + 4 =0 and 6x -8y -7 =0 are tangent to the same circle the radius of this circle is

  1. 1

  2. 2

  3. 1/4

  4. 3/4
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D 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 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.

AI explanation

The distance between the two parallel lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 can be found by rewriting the second line as 3x - 4y - 3.5 = 0. Using the distance formula for parallel lines Ax + By + C1 = 0 and Ax + By + C2 = 0, the distance is |C1 - C2| / sqrt(A^2 + B^2), which gives |4 - (-3.5)| / sqrt(3^2 + (-4)^2) = 7.5 / 5 = 1.5. If this distance is the diameter of the circle, the radius is 1.5 / 2 = 0.75 or 3/4. The radius of the circle is 3/4.

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