# Radical Line Center And Plane

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## Radical Line Center And Plane

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3. To show that the locus of points whose powers with respect to two spheres are equal is a plane perpendicular to the line joining their centres.The powers of the point P(x, y, z) with respect to the spheresS1=x2+y2+z2+2u1x+2v1y+2w1z+d1=0 S2=x2+y2+z2+2u2x+2v2y+2w2z+d2=0arex2+y2+z2+2u1x+2v1y+2w1z+d1andx2+y2+z2+2u2x+2v2y+2w2z+d2respectively

4. Equating this we obtain2x(u1-u2)+2y(v1-v2)+2z(w1-w2)+(d1-d2)=0 which is the required locus, and being of the first degree in (x, y, z), it represents a plane which is obviously perpendicular to the line joining the centres of the two spheres and is called the radical plane of the two spheres. Thus the radical plane of the two spheresS1-S2=0In case the two spheres intersect, the plane of their commoncircle is their radical plane.