<p>Write both equations with coefficients \(1\) on \(x^2,y^2\), then subtract. For distinct centres this gives a line through any common points.</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\((D_1-D_2)x+(E_1-E_2)y+(F_1-F_2)=0\)</p></div></div>
<p>Substitute the line into either circle equation. Solve the quadratic, then find both coordinates of each intersection.</p>
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<p>Two, one or no real roots correspond to two intersections, tangency or no intersection. For concentric circles, compare their radii directly.</p>

Write both equations with coefficients \(1\) on \(x^2,y^2\), then subtract. For distinct centres this gives a line through any common points.

Substitute the line into either circle equation. Solve the quadratic, then find both coordinates of each intersection.

Two, one or no real roots correspond to two intersections, tangency or no intersection. For concentric circles, compare their radii directly.

Intersection Points of Two Circles

Intersection Points of Two Circles