The given image asks to find the equation of the ellipsoid passing through specific points: $(\pm 6, 0, 0), (0, \pm 7, 0), (0, 0, \pm 1)$.

The general equation of an ellipsoid centered at the origin is:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$

Where:

• $a$ is the semi-axis length along the $x$-axis,
• $b$ is the semi-axis length along the $y$-axis,
• $c$ is the semi-axis length along the $z$-axis.

From the points provided:

• $(\pm 6, 0, 0)$ means the ellipsoid intersects the $x$-axis at $x = \pm 6$, so $a = 6$,
• $(0, \pm 7, 0)$ means the ellipsoid intersects the $y$-axis at $y = \pm 7$, so $b = 7$,
• $(0, 0, \pm 1)$ means the ellipsoid intersects the $z$-axis at $z = \pm 1$, so $c = 1$.

Thus, the equation of the ellipsoid is:

$\frac{x^2}{6^2} + \frac{y^2}{7^2} + \frac{z^2}{1^2} = 1$

Or more explicitly:

$\frac{x^2}{36} + \frac{y^2}{49} + z^2 = 1$

Would you like more details or explanations on how the ellipsoid equation is derived? Here are some related questions:

1. How do you derive the equation of an ellipsoid from its properties?
2. What is the difference between an ellipsoid and a sphere?
3. How does an ellipsoid's equation change if the center is not at the origin?
4. What are some real-world applications of ellipsoids?
5. How can we find the volume of an ellipsoid?

Tip: In a general quadratic form of a conic section or surface, constants like $a^2$, $b^2$, and $c^2$ represent squared distances from the center to the surface along their respective axes.