Multiple choice

Consider the circle $S: x^2 + y^2 - 4x - 1 = 0$ and the line $L : y = 3x - 1$. If the line $1$ cuts the circle at $A$ & $B$, then the length of the chord $AB$ is,

  1. $2\sqrt{5}$
  2. $\sqrt{5}$
  3. $5\sqrt{2}$
  4. $\sqrt{10}$
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D Correct answer
Explanation

The circle has center (2, 0) and radius sqrt(5). The perpendicular distance from the center to 3x - y - 1 = 0 is 5/sqrt(10), and the chord length is 2sqrt(5 - 25/10) = sqrt(10).

AI explanation

The center of the circle is at (2, 0) and its radius squared is 5. Substituting the line y = 3x - 1 into the circle's equation gives the quadratic 10x squared - 10x + 4 = 0. The length of the chord AB is found using the formula root of (1 plus m squared) multiplied by the root of (sum of roots squared minus 4 times product of roots), where m is 3. This evaluates to the square root of 10 multiplied by the square root of 1, giving the length of AB as the square root of 10.

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