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Given a point outside the ellipse, two tangent lines to the ellipse through that point can be constructed.
It can be proven that the angle between one tangent and the line from the point through one focus equals the angle between the other tangent and the line from the point through the other focus.
The proof would be completed by making use of the optical property of the ellipse, viz. a line from one focus is reflected to pass through the other, and they make the same angle with the tangent through the point of reflection.
The repercussion, you would see that the reflection of the starting line coincides with the other line tangent to the smaller ellipse, solving this problem. Proof for the hyperbolic case is similar by considering the construction of two tangents to the hyperbola from a given point.
So, you may wanna start thinking about constructing two tangents to an ellipse from a given point, verifing the angle property, and proving the optical property of the ellipse. Incidentally, the optical property of the ellipse can be proven geometrically by using the facts that the ellipse is convex*, and that the shortest distance between two points in flat space is a straight line**.
*(Convexity of the ellipse can be proven using the definition of the ellipse AX+XB=L, where A and B are the foci, X is a point on the ellipse and L is the major axis, as well as the definition of convexity, i.e. if X and Y are two points on the a simple closed curve, then the line segment XY lies in the interior.)
**(Of course in non-Euclidean geometry, a geodesic or the shortest path between two points is not a necessarily a straight line. In fact, on some curved surfaces, there is no such thing as a straight line. Take the sphere.)
If no one puts up those solutions, well...I would do the hard work and write them out then... |
