Multiple choice

If the lines $3x+4y-14=0$ and $6x+8y+7=0$ are both tangents to a circle, then its radius is

  1. $7$
  2. $\dfrac{7}{2}$
  3. $\dfrac{7}{4}$
  4. $\dfrac{7}{6}$
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C Correct answer
Explanation

The distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0 is |c1 - c2| / sqrt(a^2 + b^2). Here, 6x + 8y + 7 = 0 is 3x + 4y + 3.5 = 0. Distance = | -14 - 3.5 | / sqrt(3^2 + 4^2) = 17.5 / 5 = 3.5. The radius is half the distance between tangents, so 3.5 / 2 = 1.75 or 7/4.

AI explanation

The two given lines, 3x + 4y - 14 = 0 and 6x + 8y + 7 = 0, have proportional coefficients for x and y, confirming they are parallel. The distance between these parallel lines is calculated as |c2 - c1| / root(a^2 + b^2), but first, we must make their coefficients identical by dividing the second equation by 2 to get 3x + 4y + 3.5 = 0. The perpendicular distance between them is |3.5 - (-14)| / root(3^2 + 4^2) = 17.5 / 5 = 3.5. Since both lines are tangents to the same circle, the diameter of the circle equals this distance, making the radius half of 3.5, which is 7/4.

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