Question

Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5,0)

Answer 1

Given that, Length of major axis $= 26$ ; foci $= ( ±5, 0)$
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form $\dfrac{x^2}{a^2} \+ \dfrac{y^2}{b^2} = 1$, where a is the semi-major axis.
Accordingly, $2a = 26$
$⇒ a = 13$ and $c = 5.$

It is known that

$a^2 = b^2 \+ c^2.$

$13^2 = b^2+5^2$

$169 = b^2 \+ 25$

$b^2 = 169 – 25$

$b = √144$

$= 12$ (Ans.)

Thus, the equation of the ellipse is $\dfrac{x^2}{13^2} \+ \dfrac{y^2}{12^2} = 1$ or $\dfrac{x^2}{169} \+ \dfrac{y^2}{144} = 1$ (Ans.)