Given:
The equation is 49y2 – 16x2 = 784.
Let us divide the whole equation by 784, we get
\( \dfrac{49y^2}{784} – \dfrac{16x^2}{784} = \dfrac{784}{784 } \)
\(\dfrac{ y^2}{16} –\dfrac{ x^2}{49} = 1\)
On comparing this equation with the standard equation of hyperbola \( \dfrac{ y^2}{a^2} – \dfrac{x^2}{b^2} = 1\)
We get a = 4 and b = 7,
It is know that, a2 + b2 = c2
So,
c2 = 16 + 49
c2 = 65
c = \( \sqrt{65}\)
Then,
The coordinates of the foci are (0, \( \sqrt{65}\)) and (0,\( - \sqrt{65}\)).
The coordinates of the vertices are (0, 4) and (0, -4).
Eccentricity, e = \( \dfrac{c}{a} \) = \( \dfrac{ \sqrt{65}}{4}\)
Length of latus rectum = \(\dfrac{ 2b^2}{a} =\dfrac{ (2 × 7^2)}{4}\)
= \( \dfrac{(2×49)}{4} =\dfrac{ 49}{2}\)
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