Multiple choice

For a circle ${ x }^{ 2 }+{ y }^{ 2 }=81$, what is the equation of chord whose mid-point is $\left( -2,3 \right) $?

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

The equation of a chord with midpoint (x1, y1) for circle x^2 + y^2 = r^2 is xx1 + yy1 = x1^2 + y1^2. Here, x1 = -2, y1 = 3, r^2 = 81. So, -2x + 3y = (-2)^2 + 3^2 = 4 + 9 = 13. Rearranging gives 2x - 3y + 13 = 0.

AI explanation

The equation of the chord of a circle with mid-point (a, b) is given by T equals S1, where the circle is x squared plus y squared equals 81. Substituting the mid-point (-2, 3) into the formula x times a plus y times b equals a squared plus b squared gives x times -2 plus y times 3 equals -2 squared plus 3 squared. Simplifying this yields -2x plus 3y equals 13, which can be rewritten as 2x minus 3y plus 13 equals 0. This matches the required equation of the chord.

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