Question

The line $\frac{x}{k}=\frac{y}{2}=\frac{z}{- 12}$ form an isosceles triangle with the planes $2x+y+3z-1=0$ and $x+2y-3z-1=0$ , then the value of $k$ is equal to

Answer 1

If given line makes an isosceles triangle with the planes, then the angle that the line makes with the normal vectors to the planes are equal
A vector parallel to the line is $\overset{ \rightarrow }{b}= < k,2,-12>$
A normal vector to the first plane $\overset{ \rightarrow }{n_{1}}= < 2,1,3>$
A normal vector to the second plane $\overset{ \rightarrow }{n_{2}}= < 1,2,-3>$
$cos \theta = \frac{\overset{ \rightarrow }{b} \cdot \overset{ \rightarrow }{n_{i}}}{\left|\overset{ \rightarrow }{b}\right| \left|\overset{ \rightarrow }{n_{1}}\right|} = \frac{\overset{ \rightarrow }{b} \cdot \overset{ \rightarrow }{n_{2}}}{\left|\overset{ \rightarrow }{b}\right| \left|\overset{ \rightarrow }{n_{2}}\right|} \Rightarrow 2 k + 2 - 36 = k + 4 + 36$
$\Rightarrow k=74$