Multiple choice

Center of the circle inscribing a rectangle formed by the lines $x^2-8x+12=0$ and $y^2-14y+40=0$ is

  1. $(4,7)$
  2. $(7,4)$
  3. $(9,4)$
  4. $(4,9)$
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A Correct answer
Explanation

x^2-8x+12=0 => (x-6)(x-2)=0. x=2, 6. y^2-14y+40=0 => (y-10)(y-4)=0. y=4, 10. The rectangle vertices are (2,4), (6,4), (6,10), (2,10). The center is the midpoint of the diagonals: ((2+6)/2, (4+10)/2) = (4,7).

AI explanation

The lines forming the sides of the rectangle are found by solving the quadratic equations. Factoring x^2 - 8x + 12 = 0 gives x = 6 and x = 2, and factoring y^2 - 14y + 40 = 0 gives y = 5 and y = 8. These lines form a rectangle with vertices (2, 5), (6, 5), (6, 8), and (2, 8). The circle inscribing this rectangle has its center exactly at the midpoint of the diagonals, which is found by averaging the x and y coordinates of the vertices. Averaging 2 and 6 gives 4, and averaging 5 and 8 gives 7, resulting in the center (4, 7).

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