Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas 16x2 – 9y2 = 576

Asked by Pragya Singh | 11 months ago |  127

##### Solution :-

The equation is 16x2 – 9y2 = 576

Let us divide the whole equation by 576, we get

$$\dfrac{16x^2}{576}- \dfrac{9y^2}{576}$$

$$\dfrac{576}{576}$$

$$\dfrac{x^2}{36}- \dfrac{y^2}{64}=1$$

On comparing this equation with the standard equation of hyperbola

$$\dfrac{x^2}{a^2}$$ +$$\dfrac{y^2}{b^2}$$ = 1,

We get a = 6 and b = 8,

It is known that, a2 + b2 = c2

So,

c2 = 36 + 64

c2 = $$\sqrt{100}$$

c = 10

Then,

The coordinates of the foci are (10, 0) and (-10, 0).

The coordinates of the vertices are (6, 0) and (-6, 0).

Eccentricity, e = $$\dfrac{c}{a}$$

$$\dfrac{10}{6}$$ =$$\dfrac{5}{3}$$

Length of latus rectum = $$\dfrac{2b^2}{a}$$

= $$\dfrac{(2×8^2)}{6}$$

$$\dfrac{(2×64)}{6}$$

$$\dfrac{64}{3}$$

Answered by Abhisek | 11 months ago

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