Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

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ANALYTIC GEOMETRY

We now find the conjugate diameters which are equally
inclined to the axes. It is evident, geometrically, that these
will also be the conjugate diameters of equal lengths. If they
exist, their slopes must be equal except for sign: ' = - X.
Hence, by (2),
X2 b2 and X = b
a2 a
We see, then, that there is a single pair of conjugate diameters
which are equally inclined to the axes, or have equal lengths.
They are the diagonals of the rectangle circumscribed about
the ellipse (Fig. 3). We denote
D, v _ ^ B'. them by D1 and D1'.
iX. \.X. Now let a diameter D, starting
A( s\_____ % ______ from coincidence with the trans<- — "" —~T Xverse axis AA', rotate about 0
into coincidence with D1; then
^/ / l~ B \ s ~the conjugate diameter, D', starts
FIG 3 from coincidence with the conFIG. 3
jugate axis BB', and rotates into
coincidence with D1'. But D' rotates more quickly than D,*
so that the angle from D to D', at first 90~, becomes obtuse
and steadily increases. When D continues to rotate from D1
to BB', then D' rotates from D/' to AA'; but now D' rotates
less quickly than D, so that the angle from D to D' decreases
and becomes again 90~ in the final position.
EXERCISES
1. Draw accurately an ellipse whose axes are 10 cm. and
7 cm. Construct the axes and the pair of conjugate diameters
equally inclined to the axes. Then draw the diameters in* Since the slope of D1 is b < 1, f At OD1 < 45~ and > B OD1' > 45~;
a
hence D has a smaller angle through which to rotate than D'. Consequently, it is to be expected that D' will rotate more quickly than D.
A proof of the fact may easily be given later, when the student studies
the Calculus.