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

## Angle of Intersection of Two Spheres

Category : JEE Main & Advanced

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 $\frac{{{r}_{1}}{{r}_{2}}}{\sqrt{r_{1}^{2}+r_{2}^{2}}}$.

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