Question

How do you find all the critical points to graph `x^2/9 + y^2/25 = 1` including vertices, foci and asymptotes?

Answer 1

Vertices are `(-5,0)` and `(5,0)` and focii are `(0,-4)` and `(0,4)`.

Explanation:

The given equation, `x^2/9+y^2/25=1` or `x^2/3^2+y^2/5^2=1`,

is clearly the standard form of equation of an ellipse with center at `(0,0)`, major axis as `5xx2=10` (at `y`-axis) and minor axis as `3xx2=6` (at `x`-axis). Clearly vertices are `(-5,0)` and `(5,0)`.

Eccentricity is `e=sqrt(1-3^2/5^2)=sqrt(1-9/25)=sqrt(16/25)=4/5=0.8`.

And hence focii are `(0,-be)` and `(0,be)` i.e. `(0,-4)` and `(0,4)`.

As it is an ellipse question of asymptote does not arise

The graph appears as below.

graph{x^2/9+y^2/25=1 [-10, 10, -5, 5]}