Multiple choice

The co-ordinates of the middle point of the chord cut-off by $2x-5y+18=0$ by the circle $x^{2} + y^{2} -6x+2y-54=0$ are

  1. $(1,4)$
  2. $(2,4)$
  3. $(4,1)$
  4. $(1,1)$
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A Correct answer
Explanation

The center of the circle x^2 + y^2 - 6x + 2y - 54 = 0 is (3, -1). The midpoint of a chord is the projection of the center onto the line 2x - 5y + 18 = 0. Using the formula (x-3)/2 = (y+1)/(-5) = -(2(3) - 5(-1) + 18) / (2^2 + (-5)^2), we get (x-3)/2 = (y+1)/(-5) = -29/29 = -1. So x-3 = -2 => x=1, y+1 = 5 => y=4. Midpoint is (1, 4).

AI explanation

The center of the circle x squared plus y squared minus 6x plus 2y minus 54 equals 0 is (3, -1). The midpoint of the chord cut off by the line 2x minus 5y plus 18 equals 0 is simply the foot of the perpendicular from the center to this line. Using the perpendicular foot formula, we calculate the distance factor as (2*3 - 5*(-1) + 18) / (2 squared + (-5) squared) = 29/29 = 1. Subtracting the products of this factor with the coefficients from the center coordinates gives x = 3 - (1)(2) = 1 and y = -1 - (1)(-5) = 4.

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