Find the equation of the ellipsoid passing through the points (+8, 0, 0), (0₁ +3,0) and (0,0₁ +4)
Find the equation of the ellipsoid passing through the points (+8, 0, 0), (0₁ +3,0) and (0,0₁ +4)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement**
Find the equation of the ellipsoid passing through the points \( (±8, 0, 0), (0, ±9, 0), (0, 0, ±4) \).
**Solution Approach**
To determine the equation of the ellipsoid, we use the standard form:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1
\]
Given points include:
1. \( (±8, 0, 0) \) implies \( \frac{8^2}{a^2} = 1 \), hence \( a^2 = 8^2 = 64 \).
2. \( (0, ±9, 0) \) implies \( \frac{9^2}{b^2} = 1 \), hence \( b^2 = 9^2 = 81 \).
3. \( (0, 0, ±4) \) implies \( \frac{4^2}{c^2} = 1 \), hence \( c^2 = 4^2 = 16 \).
Thus, the equation of the ellipsoid is:
\[
\frac{x^2}{64} + \frac{y^2}{81} + \frac{z^2}{16} = 1
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2F437cddfd-197c-4382-a874-ed365331538c%2Fyatik2j_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find the equation of the ellipsoid passing through the points \( (±8, 0, 0), (0, ±9, 0), (0, 0, ±4) \).
**Solution Approach**
To determine the equation of the ellipsoid, we use the standard form:
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1
\]
Given points include:
1. \( (±8, 0, 0) \) implies \( \frac{8^2}{a^2} = 1 \), hence \( a^2 = 8^2 = 64 \).
2. \( (0, ±9, 0) \) implies \( \frac{9^2}{b^2} = 1 \), hence \( b^2 = 9^2 = 81 \).
3. \( (0, 0, ±4) \) implies \( \frac{4^2}{c^2} = 1 \), hence \( c^2 = 4^2 = 16 \).
Thus, the equation of the ellipsoid is:
\[
\frac{x^2}{64} + \frac{y^2}{81} + \frac{z^2}{16} = 1
\]
Transcribed Image Text:**Problem Statement:**
Find the equation of the ellipsoid passing through the points \((\pm 8, 0, 0)\), \( (0, \pm 3, 0) \), and \((0, 0, \pm 4)\).
Expert Solution
Step 1
We need to find the equation of ellipsoid passing through the points
We know that an ellipsoid has the general equation, where and are the principal semiaxes.
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