Determine algebraically whether the graph of the equation 4x^2+y^2=36 has line symmetry or symmetry wrt the origin, both, or none of these.

Hello!

The equation is

`4x^2 + y^2 = 36.`

The central (point) symmetry with respect to the origin moves a point with the coordinates (x, y) to the point (-x, -y). If a point (x_1, y_1) satisfies the equation, i.e.

`4x_1^2 + y_1^2 = 36,`

then its image `(-x_1, -y_1)` satisfies it, too:

`4*(-x_1)^2+(-y_1)^2 = 4x_1^2 + y_1^2 = 36.`

This means that this graph has central symmetry and its center of symmetry is the point (0, 0).

The same idea works for the line (reflection) symmetry. Reflection over the x-axis moves (x, y) to (x, -y), and if

`4x_2^2 + y_2^2 = 36,`

then also

`4x_2^2+(-y_2)^2=4x_2^2+y_2^2=36.`

Similarly the reflection over the y-axis moves (x, y) to (-x, y) and it is also a symmetry of this graph.

The answer: yes, this graph has line symmetry with respect to the x-axis, and with respect to the y-axis, and it has point symmetry with respect to the origin.

This graph (ellipse, actually) has no more axes of symmetry.