Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas $$\dfrac{ y^2}{9} – \dfrac{x^2}{27} = 1$$

Asked by Aaryan | 1 year ago |  72

##### Solution :-

Given:

The equation is $$\dfrac{y^2}{9} – \dfrac{x^2}{27} = 1$$

or $$\dfrac{y^2}{3^2} – \dfrac{x^2}{27^2} = 1$$

On comparing this equation with the standard equation of hyperbola $$\dfrac{ y^2}{a^2} – \dfrac{x^2}{b^2} = 1,$$

We get a = 3 and b = $$\sqrt{27}$$,

It is known that, a2 + b2 = c2

So,

c2 = $$3^2+( \sqrt{27})^2$$

= 9 + 27

c2 = 36

c = $$\sqrt{36}$$

= 6

Then,

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

The coordinates of the vertices are (0, 3) and (0, – 3).

Eccentricity, e = $$\dfrac{c}{a}= \dfrac{6}{3}=2$$

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

$$\dfrac{( 2 × 27)}{3} = \dfrac{(54)}{3} = 18$$

Answered by Aaryan | 1 year ago

### Related Questions

#### An equilateral triangle is inscribed in the parabola y2 = 4ax,

An equilateral triangle is inscribed in the parabola y2 = 4ax, where one vertex is at the vertex of the parabola. Find the length of the side of the triangle.

#### A man running a racecourse notes that the sum of the distances from the two flag posts from him is always

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always 10 m and the distance between the flag posts is 8 m. Find the equation of the posts traced by the man.

#### Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends

Find the area of the triangle formed by the lines joining the vertex of the parabola x2 = 12y to the ends of its latus rectum.