Answer: The required equation of the ellipse in standard form is [tex]\dfrac{y^2}{25}+\dfrac{x^2}{16}=1.[/tex]
Step-by-step explanation: We are given to find the equation of an ellipse in standard form with the vertical major axis of length 10 units and minor axis of length 8 units.
Since the major axis is vertical, so it will lie on the Y-axis. Let the standard form of the ellipse be given by
[tex]\dfrac{y^2}{a^2}+\dfrac{x^2}{b^2}=1,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]
where the length of major axis is 2a units and length of minor axis is 2b units.
According to the given information, we have
[tex]2a=10\\\\\Rightarrow a=\dfrac{10}{2}\\\\\Rightarrow a=5[/tex]
and
[tex]2b=8\\\\\Rightarrow b=\dfrac{8}{2}\\\\\Rightarrow b=4[/tex]
Substituting the values of a and b in equation (i), we get
[tex]\dfrac{y^2}{5^2}+\dfrac{x^2}{4^2}=1\\\\\\\Rightarrow \dfrac{y^2}{25}+\dfrac{x^2}{16}=1.[/tex]
Thus, the required equation of the ellipse in standard form is [tex]\dfrac{y^2}{25}+\dfrac{x^2}{16}=1.[/tex]
