**Category : **JEE Main & Advanced

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

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**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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