Multiple choice

A circle with centre at (2, 4) is such that the line $x+ y + 2 =0$ cuts a chord of length 6. Find the radius of the circle.

  1. $\sqrt{41}$
  2. $\sqrt{11}$
  3. $\sqrt{21}$
  4. $\sqrt{31}$
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A Correct answer
Explanation

The distance d from the center (2, 4) to the line x + y + 2 = 0 is |2 + 4 + 2| / sqrt(1^2 + 1^2) = 8 / sqrt(2) = 4 * sqrt(2). The radius r, distance d, and half-chord length a = 3 form a right triangle: r^2 = d^2 + a^2 = (4 * sqrt(2))^2 + 3^2 = 32 + 9 = 41. Thus r = sqrt(41).

AI explanation

The perpendicular distance from the center (2, 4) to the chord x + y + 2 = 0 is calculated as d = |2 + 4 + 2| / root(1^2 + 1^2) = 8 / root 2. The length of the chord is given as 6, meaning half the chord length is 3. Applying the Pythagorean theorem formula r^2 = d^2 + (L/2)^2, we get r^2 = (8 / root 2)^2 + 3^2 = 32 + 9 = 41. Therefore, the radius of the circle is root 41.

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