A hyperbola has an equation 4𝑦² − 9𝑥² + 18𝑥 − 45 = 0

Question

1. Determine the equivalent standard form of equation of a hyperbola

2. What are the coordinates of the center?

3. Find the length of the transverse axis.

4. A vertex is at,

5. An asymptote has an equation,

Graph it

Answer 1

equation should be y²/3²-(x-1)²/2²=1

center at (1,0)

vertex at x=1 y=3 and x=1, y=-3

transverse axis = 6

asymptote lines y=+/- (3/2)(x-1)

I like geogebra online graph tool, but there are others.

Answer 2

4y^2 - 9x^2 + 18x = 45

4y^2 -9(x^2 -2x + 1) = 45-9 =36

y^2/3^2 - (x-1)^2/2^2 = 1

vertical hyperbola with center at (1,0)

and vertices (1,+/-3) or (1,3), (1,-3)

asymptotes y=+/- 3/2(x-1)

transverse axis is the line x=0, the y axis

distance between the foci along the transverse axis = 2sqr13

foci are (+/-c,0) where c^2=a^2+b^2 = 9+4=13

c=sqr13,

Try desmos graphing calculator on line

It's very roughly two branches where one is U shaped and the branch below is an upside down U shape, with the U and upside down U's flattened out, approaching the straight line asymptotes