Explanation

(a)
Let the foot of the perpendicular from the origin on the given plane be $P(\alpha,\ \beta,\ \gamma)$. Since the plane passes through $A(a,\ b,\ c)$
$AP\bot OP\ \Rightarrow \ \overrightarrow{AP}.\overrightarrow{OP} = 0$
$\Rightarrow \left\lbrack (\alpha - a)\widehat{i} + (\beta - b)\widehat{j} + (\gamma - c)\widehat{k} \right\rbrack.\left( \alpha\widehat{i} + \beta\widehat{j} + \gamma\widehat{k} \right) = 0$
$\Rightarrow \ \alpha(\alpha - a) + \beta(\beta - b) + \gamma(\gamma - c) = 0$
Hence, the locus of $(\alpha,\ \beta,\ \gamma)$ is
$x(x - a) + y(y - b) + \mathcal{z}\left( \mathcal{z} - c \right) = 0$
$x^{2} + y^{2} + \mathcal{z}^{2} - ax - by - c\mathcal{z} = 0$
Which is a sphere of radius $\frac{1}{2}\sqrt{a^{2} + b^{2} + c^{2}}$