Mathematics

Mathematics

Problem

Find the coordinates of the limiting points of the system of circles determined by the two circlesx2+y2+5x+y+4=0andx2+y2+10x−4y−1=0x^2 + y^2 + 5x + y + 4 = 0 \quad \text{and} \quad x^2 + y^2 + 10x - 4y - 1 = 0
Intuitive explanation

A system of circles through two fixed circles contains a special pair of points—the limiting points—where circles in the family shrink to zero radius. The key is to use the radical axis (the locus of equal power) and recognize that limiting points lie on it and satisfy a degenerate circle equation. Subtract one circle equation from the other to find the radical axis, then use the condition that a point on this axis has zero power with respect to any circle in the family.

Intuitive explanation

A system of circles through two fixed circles contains a special pair of points—the limiting points—where circles in the family shrink to zero radius. The key is to use the radical axis (the locus of equal power) and recognize that limiting points lie on it and satisfy a degenerate circle equation. Subtract one circle equation from the other to find the radical axis, then use the condition that a point on this axis has zero power with respect to any circle in the family.

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