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LammettHash
Letting [tex]x=t[/tex], we get [tex]y=3x^2=3t^2[/tex] and [tex]z=7x^2+y^2=7t^2+(3t^2)^2=7t^2+9t^4[/tex], so we can parameterize the intersection by

[tex]\mathbf r(t)=\langle t,3t^2,7t^2+9t^4\rangle[/tex]

where [tex]-\infty<t<\infty[/tex].

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MrRoyal

A vector function r(t) is represented as: [tex]r(t) = <x,y,z>[/tex]

The vector function r(t) that represents the intersection of the two surfaces is  [tex]r(t) = <t,3t^,7t^2 + 9t^4>[/tex]

Given

[tex]z = 7x^2+ y^2[/tex]

[tex]y = 3x^2[/tex]

Let

[tex]x = t[/tex]

Substitute [tex]x = t[/tex] in [tex]y = 3x^2[/tex] and [tex]z = 7x^2+ y^2[/tex]

[tex]y = 3t^2[/tex]

[tex]z = 7t^2 + y^2[/tex]

Substitute [tex]y = 3t^2[/tex] in [tex]z = 7t^2 + y^2[/tex]

[tex]z = 7t^2 +(3t^2)^2[/tex]

[tex]z = 7t^2 +9t^4[/tex]

Recall that:

[tex]r(t) = <x,y,z>[/tex]

Substitute the values of x, y and z.

[tex]r(t) = <t,3t^,7t^2 + 9t^4>[/tex]

Hence, the vector function r(t) is:

[tex]r(t) = <t,3t^,7t^2 + 9t^4>[/tex]

Read more about vector functions at:

https://brainly.com/question/4515610

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