Question

The generalized equation of the sphere touching the three coordinate planes is

• A. $\sum x^2+2a(x+y+z)+2a^2=0$
• B. $\sum x^2-2a(x+y+z)+2a^2=0$
• C. $\sum x^2\pm 2a(x+y+z)+2a^2=0$
• D. $\sum x^2\pm 2ax\pm 2ay\pm 2az+2a^2=0$

Answer 1

The correct option is D $\sum x^2\pm 2ax\pm 2ay\pm 2az+2a^2=0$

Let the radius of the sphere be $a$; then the distance of its center from coordinate planes which it is touching should be equal to radius $a$.

Hence, its center is $(a,a,a)$.

But since the center can be in any octant we say that its center is $(\pm a,\pm a,\pm a)$ and radius $a$, so that its equation is

${(x\pm a)}^2+{(y\pm a)}^2+{(z\pm a)}^2=a^2$

$\Rightarrow x^2+y^2+z^2\pm 2ax\pm 2ay\pm 2az+2a^2=0$

There can be an infinite number of such spheres depending on the value of $a$

In case the radius $i.e.$ $a$ be fixed, then only eight such sphere can be drawn.