314 10. EUCLIDEAN GEOMETRY
10.19. When the equation y^2 = Cx - kx^2 is converted to the standard form
a2 + b^2 '
what are the quantities h, a, and b in terms of C and fc?
10.20. Show from Apollonius' definition of the foci that the product of the distances
from each focus to the ends of the major axis of an ellipse equals the square on half
of the minor axis.
10.21. We have seen that the three- and four-line locus problems have conic sections
as their solutions. State and solve the two-line locus problem. You may use modern
analytic geometry and assume that the two lines are the χ axis and the line y = ax.
The locus is the set of points whose distances to these two lines have a given ratio.
What curve is this?
10.22. Show that the apparent generality of Apollonius' statement of the three-line
locus problem, in which arbitrary angles can be prescribed at which lines are drawn
from the locus to the fixed lines, is illusory. (To do this, show that the ratio of a
line from a point Ñ to line / making a fixed angle è with the line / bears a constant
ratio to the line segment from Ñ perpendicular to I. Hence if the problem is solved
for all ratios in the special case when lines are drawn from the locus perpendicular
to the given lines, then it is solved for all ratios in any case.)
10.23. Show that the line segment from a point Ñ = (x,y) to a line ax + by = c
making angle è with the line has length
\ax + by — cj
Use this expression and three given lines : á*÷ + hy = Ci, i = 1,2,3, to formulate
the three-line locus problem analytically as a quadratic equation in two variables by
setting the square of the distance from (i, y) to line l\ equal to a constant multiple
of the product of the distances to fa and I3. Show that the locus passes through
the intersection of the line li with I2 and ^3, but not through the intersection of I2
with i 3. Also show that its tangent line where it intersects Zj is Z* itself, i = 2,3.
10.24. One reason for doubting Cavalieri's principle is that it breaks down in one
dimension. Consider, for instance, that every section of a right triangle parallel to
one of its legs meets the other leg and the hypotenuse in congruent figures (a single
point in each case). Yet the other leg and the hypotenuse are obviously of different
lengths. Is there a way of redefining "sections" for one-dimensional figures so that
Cavalieri's principle can be retained? If you could do this, would your confidence
in the validity of the principle be restored?
10.25. We know that interest in conic sections arose because of their application
to the problem of two mean proportionals (doubling the cube). Why do you think
interest in them was sustained to the extent that caused Euclid, Aristaeus, and
Apollonius to write treatises developing their properties in such detail?
10.26. Pappus' history of the conies implies that people knew that the ellipse, for
example, could be obtained by cutting a right-angled cone with a plane. Can every
ellipse be obtained by cutting a right-angled cone with a plane? Prove that it can, by
showing that any a and b whatsoever in Eq. 2 can be obtained as the section of the
