Question

Determine whether the graph of the polar equation is symmetric with respect to the horizontal axis, the vertical axis, or the pole. (Select all that apply.) MY 2 + 3 cos(@) horizontal axis vertical axis pole none of the above

Answer 1

We have given the polar coordinates so we firstly transform it into the Cartesian coordinate.

We know in general that

$x=rcos\theta;y=rsin\theta;x^2+y^2=r^2$

So we will have now

$r=2+3cos\theta$

Will be now

$\sqrt{x^2+y^2}=2+3\frac{x}{r}$

Which will be now

x2 + yr = 2r + 3.0

x2 + y2 = 21.22 + y2 + 3.0

${x^2+y^2}-3x=2\sqrt{x^2+y^2}$

On square both sides

$({x^2+y^2}-3x)^2=4({x^2+y^2})$

To check the symmetry we take the point on the graph as

(x,y)

So

$({x^2+y^2}-3x)^2=4({x^2+y^2})$

And now if we take (x,-y) so it will be

$({x^2+(-y)^2}-3x)^2=4({x^2+(-y)^2})$

$({x^2+y^2}-3x)^2=4({x^2+y^2})$

The equation will remain same

So the graph has the symmetry about x axis (horizontal axis)

So first option horizontal axis is correct answer