Multiple choice

An equation of the chord of the circle $x^{2} + y^{2} = a^{2}$ passing through the point $(2, 3)$ farthest from the centre is

  1. $ 2x + 3y = 13$
  2. $ 3x -y = 3$
  3. $ x -2y + 4 = 0$
  4. $ x -y + 1 = 0$
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A Correct answer
Explanation

The chord farthest from the center is the one perpendicular to the radius vector at that point. The slope of the radius to (2, 3) is 3/2, so the slope of the chord is -2/3. The equation is y - 3 = -2/3(x - 2), which simplifies to 3y - 9 = -2x + 4, or 2x + 3y = 13.

AI explanation

The chord of the circle x^2 + y^2 = a^2 that passes through an external point and is farthest from the center is always perpendicular to the line joining the center to that point. The center is (0, 0) and the point is (2, 3), so the slope of the line joining them is 3/2. The slope of the required chord must be the negative reciprocal, -2/3. Using the point-slope form, the equation of the chord is y - 3 = (-2/3)(x - 2), which simplifies to 2x + 3y = 13.

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