Exploring Dandelin spheres of ellipse, hyperbola and parabola. Geogebra 5 is required. Use the slider to change the shape of the cone. You may drag the points A, D or E to change the cross section of the cone. For convenience, you may click the button to get suitable ellipse, hyperbola and parabola. The two spheres touch the cone at two circles. They also touch the cross section at L and K respectively.
Ellipse Try to explain PK + PL = QR. [Hint: the properties of tangents of a circle from an external point] Hyperbola Try to explain Abs(PK - PL) = QR. [Hint: the properties of tangents of a circle from an external point] Parabola Try to explain PL = PT, where PT is perpendicula to the intersection line of the cross-section and the plane passing through the circle. [Hint: PT is parallel to the slant edge of the cone. ]