Proof of Symmetry for x^2 + 4y^2 = 1 Curve without Graph Drawing

[hms.tech]

Homework Statement

Show that the curve is symmetrical about the x-axis (without drawing the graph)
eq of the curve is : x^2 + 4y^2 = 1

also show that the curve is symmetric about the y axis

Homework Equations

The Attempt at a Solution

To prove that the curve was symmetric abou the x axis, i made x the subject of the equation of the curve:

x=(1-4y^2)^0.5 (can be positive or negative)

Then i used simple intuition:
let a particular value of y be "k" and the corresponding value of x be "c".
by simple calculation, we can conclude that for y=-k , x will still be equal to "c"

Can anyone guide me if this proof is enough or it lacks something, for the latter case, please provide an alternative but suitable proof.

 

[SammyS]

Homework Statement

Show that the curve is symmetrical about the x-axis (without drawing the graph)
eq of the curve is : x^2 + 4y^2 = 1

also show that the curve is symmetric about the y axis

Homework Equations

The Attempt at a Solution

To prove that the curve was symmetric abou the x axis, i made x the subject of the equation of the curve:

x=(1-4y^2)^0.5 (can be positive or negative)

Then i used simple intuition:
let a particular value of y be "k" and the corresponding value of x be "c".
by simple calculation, we can conclude that for y=-k , x will still be equal to "c"

Can anyone guide me if this proof is enough or it lacks something, for the latter case, please provide an alternative but suitable proof.

Generally, simply use the original equation. If you replace y with -y, and the resulting equation is equivalent to the original equation, then the graph is symmetric w.r.t. the x-axis.

For symmetry w,r,t, the y-axis, replace x with -x, in the original, and check to see that the result is equivalent to the original.

 

[Mentallic]

It is enough proof to show that f(x)=f(-x) for symmetry about the y-axis, and f(y)=f(-y) for symmetry about the x-axis. Can you see why?
Basically, this just means you just need to show that the function doesn't change when you swap x for -x and y for -y.

edit: If I bothered to refresh the page to see if a reply was already made, we wouldn't be here right now...

 

[hms.tech]

alright, from your replies i think the method i used is correct.

So thnx guys !
cheers

 

[HallsofIvy]

Yes, correct, but do you understand that everyone was telling you that you don't have to solve for one variable?