# Derivation of the Parabola's Focal Property Using a Sphere Withing a Cone

• Aug 7th 2010, 08:48 PM
zg12
Derivation of the Parabola's Focal Property Using a Sphere Withing a Cone
KMaL - Rita Ks: Conics and Dandelin spheres

I don't understand how we conclude "The angles P*PP' and P*PD are equal to the half of the apex angle of the cone"

The proof is using AAS, correct? I see that the triangles will share a leg and a right angle, but I don't understand how we can get another angle equal to use AAS.
• Aug 8th 2010, 12:22 AM
earboth
Quote:

Originally Posted by zg12
KMaL - Rita Ks: Conics and Dandelin spheres

I don't understand how we conclude "The angles P*PP' and P*PD are equal to the half of the apex angle of the cone"

The proof is using AAS, correct? I see that the triangles will share a leg and a right angle, but I don't understand how we can get another angle equal to use AAS.

1. I've modified the right sketch a little bit by adding the axis of the cone which passes through the center of the circle k and 2 axes to indicate the horizontal plane which contains the circle k.

2. Therefore $\overline{PP^*} \parallel axis_{cone}$

3. $\angle(PP'Q) = \frac12(angle\ of\ cone's\ apex)$

4. $\angle(P'PP^*) = \angle(PP'Q)$