Question

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

Answer 1

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]}