Question

Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.

4x^2 - y^2 - 24x - 4y + 28 = 0

Answer 1

Step 1:

The hyperbola equation is .

Convert the equation in standard form of hyperbola.

Change the expressions  and  into a perfect square trinomial,

add (half the x coefficient)² and add (half the y coefficient)² to each side of the equation.

The x - term is positive then the hyperbola is horizontal.

Compare it to standard form of horizontal hyperbola is .

Where  is the center of the hyperbola.

The center of the hyperbola is , and .

Step 2:

The standard form of hyperbola is .

foci is , vertices are and asymptotes are .

The hyperbola equation is .

Foci are .

Vertices are .

Vertices are .

The asymptotes are .

Comments

Step 3:

Graph the hyperbola.

Plot the center point .

Plot the focus points and vertex points  .

Draw the asymptotes are .

Sketch the hyperbola.

Solution:

Foci are .

Vertices are .

The asymptotes are .