Find a parametric representation for the lower half of the ellipsoid $7x^2 + 2y^2 + z^2 = 1$ . (Enter your answer as a comma-separated list of equations. Let $x$ , $y$ , and $z$ be in terms of $u$ and/or $v$ .)

Question :

Find a parametric representation for the lower half of the ellipsoid $7x^2 + 2y^2 + z^2 = 1$ . (enter your answer as a comma-separated list of equations. let $x$ , $y$ , and $z$ be in terms of $u$ and/or $v$ .)

Find a parametric representation for the lower half of the ellipsoid $7x^2 + 2y^ | Doubtlet.com

Solution:

Neetesh Kumar | November 14, 2024

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Step-by-step solution:

The equation of the ellipsoid is: $7x^2 + 2y^2 + z^2 = 1$

This is the equation of an ellipsoid with different coefficients for $x$ , $y$ , and $z$ , indicating different scaling along each axis. To find a parametric representation for the lower half of the ellipsoid, we can express $x$ , $y$ , and $z$ in terms of two parameters, $u$ and $v$ .

Step 1: Set up the Parametric Form

For an ellipsoid of this form, we can use spherical coordinates:

Let $x = \sqrt{\frac{1}{7}} \sin(u) \cos(v)$
Let $y = \sqrt{\frac{1}{2}} \sin(u) \sin(v)$
Let $z = \cos(u)$

The parameter $u$ will vary from $0$ to $\pi$ to cover the entire ellipsoid. However, since we want only the lower half, we restrict $u$ to $\frac{\pi}{2} \leq u \leq \pi$ , so that $z$ will be non-positive (i.e., $z \leq 0$ ).

Step 2: Final Parametric Equations

With this setup, the parametric equations for the lower half of the ellipsoid are: $x = \sqrt{\frac{1}{7}} \sin(u) \cos(v)$ $y = \sqrt{\frac{1}{2}} \sin(u) \sin(v)$ $z = -\cos(u)$

where:

$\frac{\pi}{2} \leq u \leq \pi$
$0 \leq v < 2\pi$

Answer:

The parametric representation for the lower half of the ellipsoid is: $x = \sqrt{\frac{1}{7}} \sin(u) \cos(v), \quad y = \sqrt{\frac{1}{2}} \sin(u) \sin(v), \quad z = -\cos(u)$