Multiple choice

Which of the following is a point on the common chord of the circle $x^2+y^2+2x-3y+6=0$ and $x^2+y^2+x-8y-13=0$

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

The common chord is found by subtracting the two circle equations: (x^2+y^2+2x-3y+6) - (x^2+y^2+x-8y-13) = 0, which simplifies to x + 5y + 19 = 0. Testing (1, -4): 1 + 5(-4) + 19 = 1 - 20 + 19 = 0. This point satisfies the equation.

AI explanation

The common chord of two intersecting circles is their radical axis, found by subtracting their equations. Subtracting the second equation from the first gives x plus 5y minus 19 equals 0. To determine which option is a point on this line, substitute the values for each choice into the equation. For the point (1, -4), the calculation is 1 plus 5 times -4 minus 19, which equals 0.

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