Multiple choice

The focus of the centers of the circle such that the point (2,3) is the mid point of the chord 5x+2y=16 is:

  1. 2x-5y+11=0

  2. 2x+5-11=0

  3. 2x+5y+11=0

  4. 2x-5y-11=0
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A Correct answer
Explanation

The center of the circle (h, k) must lie on a line perpendicular to the chord 5x + 2y = 16. The slope of the chord is -5/2, so the slope of the line through the center and the midpoint (2, 3) is 2/5. The equation is y - 3 = (2/5)(x - 2), which simplifies to 5y - 15 = 2x - 4, or 2x - 5y + 11 = 0.

AI explanation

The locus of centers requires the line from the center to be perpendicular to the chord, meaning its slope must be the negative reciprocal of negative 5 halves, which is 2 fifths. Using the point slope form with the center passing through (2, 3), we write y minus 3 equals 2 fifths times x minus 2. Multiplying the entire equation by 5 gives 5y minus 15 equals 2x minus 4, which simplifies to 2x minus 5y plus 11 equals 0.

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