Multiple choice

the length of a chord of a circle $x^2+y^2 =9$ intercepted by the line $x+2y=3$ is

  1. $8$
  2. $12\sqrt5$
  3. $\dfrac{12}{\sqrt5}$
  4. $5\sqrt12$
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C Correct answer
Explanation

Distance from center (0,0) to line x+2y-3=0 is d = |0+0-3| / sqrt(1^2 + 2^2) = 3/sqrt(5). Radius r = 3. Chord length = 2 * sqrt(r^2 - d^2) = 2 * sqrt(9 - 9/5) = 2 * sqrt(36/5) = 2 * 6/sqrt(5) = 12/sqrt(5).

AI explanation

Using the perpendicular distance from a point to a line formula, d = |Ax + By + C| / sqrt(A^2 + B^2), the distance from the center (0,0) to the line x + 2y - 3 = 0 is 3 / sqrt(1^2 + 2^2) = 3/sqrt(5). The radius of the circle is 3. The half-length of the chord is found by the formula sqrt(r^2 - d^2), which is sqrt(9 - 9/5) = sqrt(36/5) = 6/sqrt(5). The total length of the chord is twice this value, resulting in 12/sqrt(5).

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