The paraboloid z = 8 − x − x2 − 2y2 intersects the plane x = 3 in a parabola. Find parametric equations in terms of t for the tangent line to this parabola at the point (3, 2, −12). (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)

When , we get the parabola

We can parameterize this parabola by

Then the tangent vector to this parabola is

We get the point (3, 2, -12) when , for which the tangent vector is

Then the line tangent to the parabola at passing through the point (3, 2, -12) has vector equation

which in parametric form is

for .

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step-by-step explanation: