To solve this, let us first of all draw a diagram that represents the situation, as shown below: 

In this figure, a section of our mirror is represented by the curve APB, and our light ray originates from the origin O and reflects from the mirror at the point P.  We note that the angle of reflection must equal the angle of incidence for reflection from any mirror. 

Now, in this diagram, we require that

From the geometry of the situation around the point P, we have 

                             (1)

To progress, we take the tangent of both sides of this geometrical condition and note that if the point P is located at the coordinates (x,y), then we have 

This last expression is a standard trigonometric identity.  Combining these expressions with our geometrical condition (1) leads to 

which we can rearrange into the form 

It is not difficult to show that the solution of this differential equation is 

where c is a constant.  This is the equation of a curve known as a parabola, with the focus of the parabola located at the origin.  The following diagram shows an example of such a parabola with c = 0.1. 

Thus, a light ray shining from the origin (more generally, the focus of the parabola) will be reflected from the parabolic mirror parallel to the x-axis.  We have shown here that the parabola is the only curve that has this property.

To generate a real mirror with this property, we would generate a surface of revolution from our parabola by rotating about the x-axis.