[Math] Finding where tangent plane of ellipsoid intersects x-axis

multivariable-calculus

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This is the work I have so far:

$f_x = 2x_0,\ \ \ x_0 = x_0 \ \ \ \ \ \ f_x = 2x_0$

$f_y = 6y_0, \ \ \ \ \ y_0 = 0 \ \ \ \ \ \ f_y = 0$

$f_z = 20z_0, \ \ \ \ \ z_0 = 0 \ \ \ \ \ \ f_z = 0 \\$

$f_x(x-x_0) + f_y(y-y_0) + f_z(z-z_0) = 0$

$2x_0(1 – x_0) = 0$

$2x_0 – 2x_0^2$

$x = 1$

Apparently, the answer is actually e) but I am unsure of what I am doing wrong.

Best Answer

It is unfortunate that the question named the x-intercept $x_0$ which appears as a symbol in your formula making reference to the point on the ellipsoid. The question might be clearer if they said the x-intercept was at $(a,0,0)$ and asked you to find $a$

you should use $P_0=(x_0, y_0, z_0)=(1,-1,1)$

so the equation of the tangent plane is $2(x-1) -6(y+1)+20(z-1)=0$ and the x-intercept is at $(14,0,0)$

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