QUESTIONS AND PROBLEMS 315
right-angled cone whose equation is y^2 = zx by the plane χ = 2a — (a^2 z/b^2 ). Then
show that by taking a = eu/(l — e^2 ), b = a\/l — e^2 , χ = w, y = v, where e = h/w,
you get Eq. 1. [Hint: Recall that e is constant in a given conic section. Also,
observe that 0 < e < 1 for a section of an acute-angle cone, since h = u;tan(0/2),
where è is the vertex angle of the cone.]
10.27. As we have seen, Apollonius was aware of the string property of ellipses,
yet he did not mention that this property could be used to draw an ellipse. Do
you think that he did not notice this fact, or did he omit to mention it because he
considered it unimportant?
10.28. Prove Proposition 54 of Book 3 of Apollonius' Conies in the special case in
which the conic is a circle and the point è is at the opposite end of the diameter
from Β (Fig. 22).
