How do you find all the critical points to graph #25x^2- 4y^2=100# including vertices, foci and asymptotes?

1 Answer
Jan 1, 2017

Answer:

The center is #(0,0)#
The vertices are#(2,0)# and #(-2,0)#
The equations of the asymptotes are #y=5/2x# and #y=-5/2x#
The foci are #F=(sqrt29,0)# and #F'=(-sqrt29,0)#

Explanation:

Let's divide throughly by #100#

#(25x^2)/100-(4y^2)/100=100/100#

#x^2/4-y^2/25=1#

This is a hyperbola whose general equation is

#(x-h)^2/a^2-(y-k)^2/b^2=1#

The center is #(h,h)=(0,0)#

The vertices are #A=(h+a,k) =(2,0)#

and #A'=(h-a,k)=(-2,0)#

The asymptotes are obtained from the equation

#x^2/4-y^2/25=0#

#(x/2+y/5)(x/2-y/5)=0#

The equations of the asymptotes are #y=5/2x# and #y=-5/2x#

We calculate #c=+-sqrt(a^2+b^2)=sqrt(4+25)=sqrt29#

The foci are #F=(h+c,k)=(sqrt29,0)# and

#F'=(h-c,k)=(-sqrt29,0)#

graph{(x^2/4-y^2/25-1)(y-5/2x)(y+5/2x)=0 [-6.24, 6.244, -3.12, 3.12]}