Multiple choice

Tangents are drawn from $(4,4)$ to the cricel ${x}^{2}+{y}^{2}-2x-2y-7=0$ to meet the circle at $A$ and $B$. The length of the chord $AB$ is

  1. $2\sqrt{3}$
  2. $3\sqrt{2}$
  3. $2\sqrt{6}$
  4. $6\sqrt{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The circle has center (1, 1) and radius 3. The distance from the center to (4, 4) is 3sqrt(2), so the distance from the center to the chord of contact is 9/(3sqrt(2)). Using the chord formula gives AB = 3sqrt(2).

AI explanation

The center of the circle is (1, 1) and its radius is the square root of (1 + 1 + 7), which is 3. The distance from the point (4, 4) to the center is the square root of (4-1)^2 + (4-1)^2, equaling the square root of 18. The length of the chord of contact AB is given by the formula 2 times the square root of (distance^2 - radius^2) times radius, divided by distance. Plugging in the values gives 2 times the square root of (18 times (18 - 9)) divided by the square root of 18, which simplifies to exactly 3 times the square root of 2.

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