Find the maximum and minimum values of $\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}$ when $lx+my+nz=0$ and $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
I tried with Lagrange's method for two conditions with the following equation
$f_1=\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}$
$g_2=lx+my+nz=0$
$g_3=\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$
Then : $\nabla f_1=\lambda_1 \nabla g_2 +\lambda_2 \nabla g_3$ and also used the constrain equations but I am not able to find the maximum and minimum