Given:
The equation is 36x2 + 4y2 = 144 or
\( \dfrac{x^2}{4} + \dfrac{y^2}{36 }= 1\)
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.
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 = \( \dfrac{ 2b^2}{a} =\dfrac{ (2×2^2)}{6}\)
= \( \dfrac{4}{3} \)
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