252 
ELLIPSE. 
Hence the equation to PD is, by the formula 
(x" - x) y - {y" -y')x = x'y - xy\ 
y (— a sin# — a cos#) — x {b cos# - b sin#) 
= — a sin #. b sin # — a cos #. b cos #, 
or (<ay — bx) sin# + (ay + bx) cos# = ab (1). 
Differentiating with respect to #, we have 
{ay — bx) cos# — {ay + bx) sin# = 0 (2). 
Squaring and adding (1) and (2), we get, for the equation to the 
required locus, 
2ay + 25V = dP\ 
2. If, from every point in the axis major of an ellipse, as 
centre, a circle be described, with radius equal to the ordinate at 
that point: to find the envelop of the circles. 
The equation to the ellipse being 
that to the envelop will be an arc of another ellipse represented 
by the equation 
x 2 y 2 
Ll = 
a 2 + b 2 + b 2 
3. Two straight lines are drawn from one extremity of the 
major axis of an ellipse, making with it angles the tangents of 
which are in a given ratio: to find the locus of the ultimate 
intersections of chords, each of which joins the two points where 
two corresponding lines cut the ellipse. 
The equation to the ellipse being 
f = ^ (2«« - x 1 ), 
the envelop required will be another ellipse, the equation of 
which referred to the same axes will be, a denoting the given 
ratio, AnH 1 
‘2 ‘2\