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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.

Find the equation of the ellipsoid passing through the points (±6, 0, 0), (0, ±7, 0), and (0, 0, ±1).

Learn how to find the equation of an ellipsoid that passes through the points (±6, 0, 0), (0, ±7, 0), and (0, 0, ±1). The detailed solution includes deriving the general form of an ellipsoid equation. This problem is suitable for advanced geometry students in Grades 11-12.

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