The distances of the point ( 1, - 5, 9 ) from the plane x + y + z =...

Given equation of plane is $x + y + z = 5$.

The distance measured along the line $x = y = z$.

Direction ratios of the given line is $( 1, 1, 1 )$.

So, the equation of line $PQ$ is

$\dfrac{x - 1}{1} = \dfrac{y + 5}{1} = \dfrac{z - 9}{1} = \lambda$

$⇒ x = \lambda + 1, y = \lambda - 5, z = \lambda + 9$

lies on the plane $x + y + z = 5$

$∴ ( \lambda + 1 ) - ( - \lambda + 5 ) + ( \lambda + 9 ) = 5$

$⇒ \lambda = - 10$

The coordinate of $Q$ is $( - 9, - 15, - 1 )$ and the coordinate of $P$ is $( 1, - 5, 9 )$

$PQ = \sqrt{( 10{)}^{2} + ( 10{)}^{2} + ( 10{)}^{2}} = 10\sqrt{3}$

$∴ 2\sqrt{3}k = 10\sqrt{3} ⇒ k = 5$