Partial Derivatives - Finding tangent in a volume?

[Shaybay92]

Not sure I understand exactly what this question is asking. This is obviously a volume in R3 and so how do you get a tangent inside a volume? Or is it just along the plane y = 2 intersecting the volume? Also, what is a parametric equation...? Thanks for the help:

Question:
The ellipsoid 4x^2 + 2y^2 + z^2 = 16 intersects the plane y = 2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1,2,2).

Attempt:
I substituted in y = 2 and got the equation 4x^2 + z^2 = 8. Do I just find what the partial derivatives are at the point (1,2)? And would the tangents be combinations of these two derivatives...?

 

[Phrak]

You want to find the tangent line to an ellipse. The ellipse lies in the plane y=2, so you don't have to worry about tangent lines in a volume. Solve for the ellipse in the XZ-plane at y=2, and find its tangent at (x,z)=(1,2). You should be able to convince yourself that the tangent to the ellipse lies in the y=2 plane. And (1,2) better be a point on this ellipse (it is) or there is something wrong with the problem statement.

 

[Shaybay92]

Thanks but I'm not so good with ellipse equations and I don't really understand, if y=2 then the ellipse would be 4x^2 + z^2 = 8 right... but this isn't an equation of two variables is it? Because its an implicit function of z (is this correct?). So solving for z would give a one variable function but then it is not partial derivatives because its only a single variable function right?

 

[Phrak]

I'm not a homework helper. I'm just filling in for a short moment, but, yes, 4x^2 + z^2 = 8 is an equation of two variables.

For the elliptical curve, all the partial derivatives with respect to y disappear. The curve is stuck in the y=2 plane. So obviously it doesn't change with y.

 

[HallsofIvy]

Yes, the ellipse, in the plane y= 2 is 4x^2+ z^2= 8. To find the equation of the tangent line at (1, 2, 2), you can either think of z as a function of x or x as a function of z.

Since "z as a function of x" seems more natural, by "implicit differentiation", 4x + 2z z'= 0 gives, at (1, 2, 2), 4+ 4z'= 0 so that z'= -1.

Find z= ax+ b for that slope and point and include y= 2. If you want parametric equations, let x= t, z= at+ b, y= 2.

 

[Redbelly98]

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