Reflection properties of the conic sections

The ellipse.

The rays coming from one of the foci of an ellipse are reflected in the direction of the other focus.

Let F and G be the focal points of an ellipse. The following applet shows a point P on the ellipse and the straight line tangent to the ellipse on P. The angle that forms the "incident ray" FP with the tangent line is equal to the angle formed by the "reflected ray" GP on the other side. 

This is the so called reflection property of the ellipse. The following applet helps to understand why the reflection property holds.

F and G are the focal points of an ellipse. P is a point of the ellipse and r is the tangent line to the ellipse in P. Gr is the reflection of G with respect to r, that is to say, GP and GrP form the same angle with the straight line r. It will be proved that FP and GP form the same angle with r.

Q is an arbitrary point of r. Therefore

FQ+QG=FQ+QGr>FP+PGr=FP+PG

unless Q=P. This means that the minimum value of the sum FQ+QGr is reached when Q=P and therefore P is in line with F and Gr. Therefore the angles of FP and PGr with the straight line r are equal, but as the second is equal to that which forms GP with r, it turns out that FP and GP form the same angle with r, which is what we wanted to demonstrate.

The hyperbola.

The rays coming from one focus of a hyperbola are reflected in such a way that the reflected rays seem to come from the other focus.

F and G are the focal points of an ellipse. The following applet shows a point P on the hyperbola and the tangent line to the hyperbola in P. The angle which forms the "incident ray" FP with the straight line is equal to that which forms the "reflected ray", which is a prolongation of the segment GP, that is to say, the reflected ray seems to come from the second focal point G.

This is the so called reflection property of the hyperbola.

The following applet shows a hyperbola with focal points F and G and a point P on it. Let Gr be a point on the straight line FP which is as far from P as G and let r be the straight line formed by the equidistant points of G and Gr. Therefore, from the properties of the sides of the triangle FQGr, it follows that for all the points Q of the straight line,

FQ-QGr>=FGr=FP-PGr

and the equality holds only when Q=P. This shows that the only point on the straight line r which touches the hyperbola is P, and therefore r is the tangent line to the hyperbola in P. From here, it follows that the angle FPr is equal to the angle rPG, which is the property of reflection of the hyperbola that we wanted to prove.

The parabola.

The rays coming from the focus of a parabola are reflected parallel to its axis of symmetry. Inversely, the rays coming from a far away source and are parallel to the axis of symmetry of a parabola, on being reflected, are concentrated in the origin. 

F is the focus of the parabola. The following applet shows a point P on the parabola and the tangent line in P. The angle that forms the "incident ray" FP with the tangent line is equal to the angle formed by the "reflected ray", which is parallel to the axis of symmetry of the parabola.

This is the so called reflection property of the parabola.

The following applet shows a parabola with the focus F and a point P on it. d is the directrix of the parabola and P´ the point of the directrix whose distance to P is minimal. From the properties of the parabola we have FP´=FP. t is the straight line of the equidistant points of F and P´. Therefore t passes through P. For any point Q of t, let Q´ be the point on d closest to Q. Therefore P´Q<=Q´Q and the equality holds only when Q=P. This shows that t only touches the parabola at the point P. Therefore t is a tangent line to the parabola. However, since the angle FPt is equal to the angle tPP´, it turns out that the reflection of FP with respect to t is perpendicular to the directrix d and therefore is parallel to the axis of symmetry of the parabola, which is what we wanted to demonstrate.

Author: José Luis Abru León

	Ministerio de Educación, Cultura y Deporte. Año 2000