Value of k so that the triangle formed becomes isosceles

Question

Let a line be $\frac {x}{k}=\frac {y}{2}=\frac {z}{-12} $. For what value of $k$ does it make an isosceles triangle with planes $2x+y+3z=1,x+2y-3z=1$. $$\text {attempt} $$ let the point on the line be $tk,2t,-12t $ so from this we have an equation which is $|2tk++2t-36t-1|=|tk+4t+36t-1|$ as the triangle is isosceles. From which other should I get the second equation to solve for $t,k $

Answer 1

I think it happens for all value of $k$.

Let $A(0,0,0)$ and let $AB$ and $AC$ be altitudes to our plains.

Thus, $AB=AC$ and $\Delta ABC$ is isosceles.