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