Answer:
[tex]k=54[/tex]
Step-by-step explanation:
Find the coordinates of points M and N in terms of k. Solve the system of two equations:
[tex]\left\{\begin{array}{l}y=3x^2\\ \\y=-3x^2+k\end{array}\right.\Rightarrow \left\{\begin{array}{l}y=3x^2\\ \\y=-y+k\end{array}\right.\\ \\2y=k\\ \\y=\dfrac{k}{2}\\ \\\dfrac{k}{2}=3x^2\\ \\x^2=\dfrac{k}{6}\\ \\x=\pm\sqrt{\dfrac{k}{6}}[/tex]
Two points have the coordinates [tex]M\left(\sqrt{\dfrac{k}{6}},\dfrac{k}{2}\right)[/tex] and [tex]N\left(-\sqrt{\dfrac{k}{6}},\dfrac{k}{2}\right)[/tex]
Find the distance between M and N:
[tex]MN=\sqrt{\left(\sqrt{\dfrac{k}{6}}-\left(-\sqrt{\dfrac{k}{6}}\right)\right)^2+\left(\dfrac{k}{2}-\dfrac{k}{2}\right)^2}=\sqrt{4\dfrac{k}{6}}=\sqrt{\dfrac{2k}{3}}[/tex]
Since MN = 6, you have
[tex]\sqrt{\dfrac{2k}{3}}=6\\ \\\dfrac{2k}{3}=36\\ \\2k=108\\ \\k=54[/tex]
