Given:
The equation of the ellipse is
[tex]3x^2+2y^2=k[/tex]
The length of its major axis is 6.
To find:
The value of k.
Solution:
The standard form of an ellipse is
[tex]\dfrac{x^2}{b^2}+\dfrac{y^2}{a^2}=1[/tex]
Where, a>b and 3a is the length of major axis.
We have,
[tex]3x^2+2y^2=k[/tex]
Divide both sides by k.
[tex]\dfrac{3x^2+2y^2}{k}=\dfrac{k}{k}[/tex]
[tex]\dfrac{x^2}{\frac{k}{3}}+\dfrac{y^2}{\frac{k}{2}}=1[/tex]
[tex]\dfrac{x^2}{(\sqrt{\frac{k}{3}})^2}+\dfrac{y^2}{(\sqrt{\frac{k}{2}})^2}=1[/tex]
Here, [tex]\sqrt{\frac{k}{2}}>\sqrt{\frac{k}{3}}[/tex].
So, the length of the major axis is
[tex]2\sqrt{\dfrac{k}{2}}=6[/tex]
[tex]\sqrt{\dfrac{k}{2}}=3[/tex]
[tex]\dfrac{k}{2}=9[/tex]
[tex]k=18[/tex]
Therefore, the value of k is 18.
