JEE Main & Advanced Mathematics Three Dimensional Geometry Angle of Intersection of Two Spheres

If the angle of intersection of two spheres is a right angle, the spheres are said to be orthogonal.

Condition for orthogonality of two spheres :

Let the equation of the two spheres be

\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2ux+2vy+2wz+d=0\]               .....(i)

and         \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2{u}'x+2{v}'y+2{w}'z+{d}'=0\]            .....(ii)

If the sphere (i) and (ii) cut orthogonally, then \[2u{u}'+2v{v}'+2w{w}'=d+{d}',\] which is the required condition.

• If the spheres \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{a}^{2}}\] and \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}\] \[+2ux+2vy+2wz+d=0\] cut orthogonally, then \[d={{a}^{2}}\].

• Two spheres of radii \[{{r}_{1}}\] and \[{{r}_{2}}\] cut orthogonally, then the radius of the common circle is \[\frac{{{r}_{1}}{{r}_{2}}}{\sqrt{r_{1}^{2}+r_{2}^{2}}}\].