7.4 Hyperbolas

Find the hyperbolas given the following conditions:

(a) The foci are [math](5,3), (5, -5)[/math] and the length of the conjugate axis is 6.

(b) The foci are [math](4,0), (-4, 0)[/math] and the asymptotes are [math]y=\pm x[/math].

Consider the hyperbolas [math]\dfrac{x^2}{4}-\dfrac{y^2}{9}=1[/math]:

(a) If the focal lengths is [math]2\sqrt{a}[/math] where [math]a[/math] is an integer, what is the value of [math]a[/math]?

(b) What is the slope of the asymptote of the hyperbolas?

Find the asymptotes of [math]\dfrac{(y+1)^2}{16}=1+(x-2)^2[/math]. Note that each space is a positive integer.

Let [math]\Gamma[/math] be a hyperbola having asymptotes [math]y=\pm2x+1[/math]. Suppose [math](3,1)[/math] is a vertex of [math]\Gamma[/math], find the equation of [math]\Gamma[/math].

Let [math]\Gamma[/math] be a hyperbola with foci [math](-2,7)[/math] and [math](-2,-3)[/math]. Suppose [math]\Gamma[/math] passes through [math](-2,5)[/math]. Find the equation of [math]\Gamma[/math].

Let [math]\Gamma_1[/math] and [math]\Gamma_2[/math] be hyperbolas having common asymptotes. Which of the following condition(s) prove(s) that [math]\Gamma_1=\Gamma_2[/math]?

Consider two points [math]A(0,5)[/math] and [math]B(0,-5)[/math]. Let [math]\Gamma[/math] be a curve satisfying [math]3PA=2PB[/math] for [math]P\in\Gamma[/math]. What is [math]\Gamma[/math]?

Consider an ellispe [math]\Gamma_1: \dfrac{(x-1)^2}{4}+\dfrac{(y+2)^2}{9}=1[/math] and a hyperbola [math]\Gamma_2[/math]. If the foci of [math]\Gamma_1[/math] are the vertices of [math]\Gamma_2[/math] and the vertices of [math]\Gamma_1[/math] are the foci of [math]\Gamma_2[/math]. Find the equation of [math]\Gamma_2[/math].

A hyperbola [math]\Gamma[/math] has asymptotes [math]4x-3y=-2[/math] and [math]4x+3y=10[/math]. Suppsoe the transverse axis of [math]\Gamma[/math] is parallel to the [math]y[/math]-axis and the focal length is 10. Find the equation of [math]\Gamma[/math].