Find a vector function that represents the curve of intersection of the two surfaces: the top half of the ellipsoid $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$ and the parabolic cylinder $y = x ^ { 2 }$.
A) $\mathrm { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 25 - t ^ { 2 } - 5 t } { 5 } } \mathbf { k }$
B) $\mathbf { r } ( t ) = t \mathbf { i } - t ^ { 2 } \mathbf { j } - \sqrt { \frac { 25 - t ^ { 2 } - 5 t } { 5 } } \mathbf { k }$
C) $\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 5 - t ^ { 2 } - 5 t ^ { 4 } } { 5 } } \mathbf { k }$
D)$\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 4 } \mathbf { j } + \sqrt { \frac { 25 - t ^ { 2 } + 5 t } { 5 } } \mathbf { k }$
E)$\mathbf { r } ( t ) = t \mathbf { i } + t ^ { 2 } \mathbf { j } + \sqrt { \frac { 25 + t ^ { 2 } - 5 t ^ { 4 } } { 5 } } \mathbf { k }$

Question

Quizplus · Accepted Answer

Substituting $y = x^2$ into the ellipsoid equation $x^2 + 5y^2 + 5z^2 = 25$ gives $x^2 + 5x^4 + 5z^2 = 25$. Solving for $z$ in terms of $x$ (or $t$) leads to $z = \sqrt{\frac{25 - x^2 - 5x^4}{5}}$, which matches the form in option A when considering $x = t$.

Find a Vector Function That Represents the Curve of Intersection x2+5y2+5z2=25x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25x2+5y2+5z2=25

Find a Vector Function That Represents the Curve of Intersection $x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } = 25$