The standard equation of an ellipsoid is given by
[tex]\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1[/tex]
where a, b and c are semi axes length. It means the ellipsoid is passing through the points (±a,0,0),(0,±b,0) and (0,0,±c).
Now, we have been given that the ellipsoid is passing through the points (±6,0,0),(0,±7,0) and (0,0,±5). Thus, we have
[tex]a=\pm 6\\ b=\pm 7\\ c=\pm 5\\[/tex]
Therefore, the equation of the ellipsoid is given by
[tex]\frac{x^2}{(\pm 6)^2}+\frac{y^2}{(\pm 7)^2}+\frac{z^2}{(\pm 5)^2}=1\\ \\ \frac{x^2}{36} +\frac{y^2}{49} +\frac{z^2}{25} =1[/tex]
The equation of the ellipsoid is [tex]\frac{x^2}{36} + \frac{y^2}{49} + \frac{z^2}{25} =1[/tex]
The equation of an ellipsoid is represented as:
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1[/tex]
Given that the ellipsoid passes through the points (±6,0,0),(0,±7,0) and (0,0,±5)
It means that:
(±a,0,0) = (±6,0,0)
(0,±b,0) = (0,±7,0)
(0,0,±c) = (0,0,±5)
By comparison, we have:
[tex]\pm a = \pm 6[/tex]
[tex]\pm b = \pm 7[/tex]
[tex]\pm c = \pm 5[/tex]
So, the equation of the ellipsoid becomes
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} =1[/tex]
[tex]\frac{x^2}{\pm 6^2} + \frac{y^2}{\pm 7^2} + \frac{z^2}{\pm 5^2} =1[/tex]
Evaluate the squares
[tex]\frac{x^2}{36} + \frac{y^2}{49} + \frac{z^2}{25} =1[/tex]
Hence, the equation of the ellipsoid is [tex]\frac{x^2}{36} + \frac{y^2}{49} + \frac{z^2}{25} =1[/tex]
Read more about equations at:
https://brainly.com/question/1559324
