Question

Find the equation of the tangent plane to the conicoid x² +y² = kz at the point (k, k,2k), where k is a constant. Represent the plane geometrically. Now take different values of k, including both positive and negative, and see how the shape of the conicoid changes.

Answer 1

We have an equation of conicoid

$$F(x,y,z) = x^2 + y^2 - kz$$

And the point

$$M(k,k,2k)$$

Let’s find a derivative

$$F'_x(x,y,z) = 2x; F'_x(M) = 2k$$ $$F'_y(x,y,z) = 2y; F'_y(M) = 2k$$ $$F'_z(x,y,z) = -k; F'_z(M) = -k$$

So equation of tangent plane is:

$$F'_x(M)*(x-k) + F'_y(M)*(y-k) + F'_z(M)*(z-2k) = 0$$ $$2k*(x-k) + 2k*(y-k)+(-k)*(z-2k) = 2kx + 2ky - kz -2k^2 = 0$$

Answer:

$$2kx + 2ky - kz -2k^2 = 0$$

if k > 0

if k < 0