List the intercepts and test for symmetry `x^2+49y^2=49`
(1) If x=0 `49y^2=49==>y^2=1==>y=+-1`
The y-intercepts are 1 and -1
(2) If y=0 `x^2=49==>x=+-7`
The x-intercepts are 7 and -7
(3) If we substitute (-x) for x we get `(-x)^2+49y^2=49 ==> x^2+49y^2=49` so the graph is symmetric across the y-axis
If we substitute (-y) for y we get `x^2+49(-y)^2=49==>x^2+49y^2=49` so the graph is symmetric across the x-axis.
Replacing x with (-x) and y with (-y) does not change the relation, so the graph is symmetric about the origin.
This is not a surprise as `x^2+49y^2=49==>x^2/49+y^2/1=1`
This is an ellipse centered at the origin.