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 Sakshi | 1 year ago |  57

##### Solution :-

Given:

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 Aaryan | 1 year ago

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