Question

Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 3x2 + 2y2 + 4z2 = 21 at the point (−1, 1, 2). (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)

Answer 1

`x=-1-18 t,y=1+19 t,z=2-2 t`

Explanation:

We are given that

Equation of paraboloid

`z=x^2+y^2`

`z-x^2-y^2=0`

And equation of the ellipsoid

`3x^2+2y^2+4z^2-21=0`

We have to find the parametric equation for tangent line to the curve of the intersection of the paraboloid and ellipsoid at point (-1,1,2).

We have to find the normal at point (-1,1,2)

`N_1=-2x\hat{i}-2y\hat{j}+\hat{k}`

Normal at point (-1,1,2)

`N_1=<2,-2,1>`

`N_2=6x\hat{i}+4y\hat{j}+8z\hat{k}`

Normal at point (-1,1,2)

`N_2=<-6,4,16>`

We rescale and set
`N_2=<-3,2,8>`

The tangent vector to the curve of intersection is given by

`N_1\time N_2`=
`\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&-2&1\\-3&2&8\end{vmatrix}`

`N_1* N_2=<-18,19,-2>`

Hence, the tangent line is given by

`x=-1-18 t,y=1+19 t,z=2-2 t`