# How do you write a standard form equation for the hyperbola with asymptotes are at y= 1/2 x and y= -1/2x. The hyperbolas vertices are at (-3,0) and (3,0)?

Jan 20, 2017

${x}^{2} / {\left(\frac{3}{2}\right)}^{2} - {y}^{2} / {\left(\frac{3}{2}\right)}^{2} = 1$. See the asymptotes-inclusive graph of the rectangular hyperbola (RH) in the Socratic graph.

#### Explanation:

If A1=9 and A2 =0 are the equations to the asymptotes, the quation

to the family of hyperbolas with thesr aymptotes is

$A 1 \times A 2 = c$. c is called the parameter.

Here, it is

(y-x/2)(y+x/2)= c

The parameter value for the member of this family that has the

vertices A( 3, 0) and A'(-3, 0) is given by

$\left(0 - \frac{3}{2}\right) \left(0 + \frac{3}{2}\right) = - \frac{9}{4} = c .$

Now, the equation is in the standard form

${x}^{2} / {\left(\frac{3}{2}\right)}^{2} - {y}^{2} / {\left(\frac{3}{2}\right)}^{2} = 1$, revealing that it is rectangular.

See the asymptotes-inclusive RH in the graph.

graph{(x^2-y^2-9/4)(x^2-y^2/4)=0 [-10, 10, -5, 5]}