Question

The equation ^2 − 4 − 4^2 + 13 = 0 will produce a hyperbola. How can we tell by simply observing the equation?In what directions do the branches of this hyperbola open? How do you know? Explain. Sketch a graph of this hyperbola, clearly indicating how you have determined thekey characteristics (center, vertices, eccentricity, foci). Give the domain and range of this hyperbola.

Answer 1

we have the equation

$^2−4−4^2+13=0$

Group similar terms and move the constant to the right side

$(^2−4)−4^2=-13$

Complete the square

$\begin{gathered} (y^2-4y+2^2-2^2)-4x^2=-13 \\ (y^2-4y+2^2)-4x^2=-13+2^2 \\ (y^2-4y+2^2)-4x^2=-9 \end{gathered}$

Rewrite as a perfect square

$(y-2)^2-4x^2=-9$

Divide both sides by -9

$\begin{gathered} \frac{(y-2)^2}{-9}-\frac{4x^2}{-9}=\frac{-9}{-9} \\ \\ -\frac{(y-2)^2}{9}+\frac{x^2}{\frac{9}{4}}=1 \\ \\ \frac{x^{2}}{\frac{9}{4}}-\frac{(y-2)^{2}}{9}=1 \\ \end{gathered}$

The coordinates of the center are (0,2)

The transverse axis is on the x-axis

a^2=9/4 -----------> a=3/2

b^2=9 -----------> b=3

The vertices are --------> (0+1.5,2) and (0-1.5,2)

so

Vertices at (1.5,2) and (-1.5,2)

Find out the value of c

c^2=a^2+b^2

c^2=(9/4)+9

c^2=45/9

c=√5

Find out the coordinates of the foci

(0+√5,2) and (0-√5,2)

using a graphing tool

The domain is the interval (-infinite, -1.5) U (1.5, infinite)

The range is the interval (-infinite, infinite)