Explanation

(a)
The equation of the plane through the line of intersection of the plane $4x + 7y + 4\mathcal{z} + 81 = 0$ and $5x + 3y + 10\mathcal{z} = 25$ is $\left( 4x + 7y + 4\mathcal{z} + 81 \right) + \lambda\left( 5x + 3y + 10\mathcal{z} - 25 \right) = 0$
$\Rightarrow (4 + 5\lambda)x + (7 + 3\lambda)y + (4 + 10\lambda)\mathcal{z} + 81 - 25\lambda = 0$ (i)
Which is perpendicular to $4x + 7y + 4\mathcal{z} + 81 = 0$
$\Rightarrow 4(4 + 5\lambda) + 7(7 + 3\lambda) + 4(4 + 10\lambda) = 0$
$\Rightarrow 81\lambda + 81 = 0$
$\Rightarrow \lambda = - 1$
Hence the place is $x - 4y + 6\mathcal{z} = 106$