Multiple choice

If the lines 3x + 4y - 14 = 0, 6x + 8y + 7 = 0 are both tangents of the same circle, its diameter is

  1. 3.2

  2. 4.5

  3. 3.5

  4. 4.2

  5. 4.6
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C Correct answer
Explanation

The lines are 3x + 4y - 14 = 0 and 6x + 8y + 7 = 0. Rewrite the second as 3x + 4y + 3.5 = 0. The distance between these parallel lines is the diameter. Distance = |c1 - c2| / sqrt(a^2 + b^2) = |-14 - 3.5| / sqrt(3^2 + 4^2) = 17.5 / 5 = 3.5.

AI explanation

First, divide the second equation by 2 to get 3x + 4y + 3.5 = 0, revealing that the two lines are parallel. The diameter of the circle caught between them is the perpendicular distance between these parallel lines, calculated as the difference of their constants divided by the root of their coefficients squared. This gives a diameter of 17.5 divided by 5, which is 3.5.

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