Question

Find  the  center,  vertices,  foci,  and  the equations of the asymptotes of the hyperbola. Use a graphing utility to graph the hyperbola and its asymptotes.

9y^2 - x^2 + 2x + 54y + 62 = 0

Answer 1

Step 1:

The equation is .

Group the terms.

Complete each square.

Compare it to the standard form of the hyperbola .

Here transverse axis is parallel to the axis.

is the center of the hyperbola.

is the distance between center and focus.

.

Step 2:

The center of the hyperbola is .

The vertices of the hyperbola are .

The foci of the hyperbola are .

Find the points to form a rectangle.

.

.

The extensions of the diagonals of the rectangle are the asymptotes of the hyperbola.

Asymptotes of the hyperbola are .

Substitute the values of in .

Asymptotes are .

Step 3:

Graph :

(1) Draw the coordinate plane.

(2) Draw the equation of the hyperbola.

(3) Plot the foci and vertices.

(4) Form a rectangle containing the points , .

(5) Draw the asymptotes of the hyperbola.

Solution :

The equation of the hyperbola is .

The center of the hyperbola is .

The vertices of the hyperbola are .

The foci of the hyperbola are .

Asymptotes of the hyperbola are .

Graph of the hyperbola and asymptotes :