- The point \(P\) has coordinates \(( 1,2,1 )\)
The line \(l\) has Cartesian equation
$$\frac { x - 3 } { 5 } = \frac { y + 1 } { 3 } = \frac { z + 5 } { - 8 }$$
The plane \(\Pi _ { 1 }\) contains the point \(P\) and the line \(l\).
- Show that a Cartesian equation for \(\Pi _ { 1 }\) is
$$6 x - 2 y + 3 z = 5$$
The point \(Q\) has coordinates \(( 2 , k , - 7 )\), where \(k\) is a constant.
- Show that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is
$$\frac { 2 } { 7 } | k + 7 |$$
The plane \(\Pi _ { 2 }\) has Cartesian equation \(8 x - 4 y + z = - 3\)
Given that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is the same as the shortest distance between \(\Pi _ { 2 }\) and \(Q\), - determine the possible values of \(k\).