The equation is 36x2 + 4y2 = 144 or
\( \dfrac{x^2}{4}\)+ \( \dfrac{y^2}{36}\) = 1
or \( \dfrac{x^2}{2^2}\) + \( \dfrac{y^2}{6^2}\)= 1
Here, the denominator of \( \dfrac{y^2}{6^2}\) is greater than the denominator of \( \dfrac{x^2}{2^2}\).
So, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with
\( \dfrac{x^2}{b^2}\) + \( \dfrac{y^2}{a^2}\)= 1, we get b = 2 and a = 6.
c = \( \sqrt{ (a^2 – b^2)}\)
= \( \sqrt{ (36– 4)}\)
= \( \sqrt{32}\)
=\( 4\sqrt{2}\)
Then,
The coordinates of the foci are (0,\( 4\sqrt{2}\)) and (0, \( - 4\sqrt{2}\)).
The coordinates of the vertices are (0, 6) and (0, -6)
Length of major axis = 2a = 2 (6) = 12
Length of minor axis = 2b = 2 (2) = 4
Eccentricity, e =\( \dfrac{c}{a}\) = \( \dfrac{ 4\sqrt{2}}{6}\) =\( \dfrac{ 2\sqrt{2}}{3}\)
Length of latus rectum = \( \dfrac{2b^2}{a}\)
= \( \dfrac{(2×2^2)}{6}\) = \( \dfrac{(2×4)}{6}\)
= \( \dfrac{4}{3}\)
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