254 
ELLIPSE. 
and 
whence 
a 
Also a + V is least when sin7 is greatest: in this case, from (1) 
and (2), 
a + V = a + &, 
a — b' — a — b, 
a = a, b' = b. 
and 
whence 
Durrande : Gergonne, Annales de Mathématiques, tom. xil. p. 168. 
2. If a straight line be drawn from the focus of an ellipse, 
the eccentricity of which is e, so as to make a given angle ¡3 
with the tangent ; to shew that the locus of its intersection with 
the tangent will be a circle, which touches or lies entirely with 
out the ellipse as cos/3 is < or > e. 
Let SY be the perpendicular from the focus S upon the 
tangent PT at P, which cuts the semi-axis major CA, produced, 
in T. Let Q be a point in PP, such that ¿.SQP= /3. 
Let ¿PTS — </>, SQ -=- r, lQST = 0. 
Then r sin/3 = 8Y= [a* sin 2 </> + P cos 2 </>)^ — ae sin</>. 
Squaring and putting for </> its value /3 — 0, we have, for the 
equation to the locus of Q, 
r 2 siP/S + 2aer sin/8 sin (/3 — 6) = a? (1 — e‘ 2 ), 
which is the polar equation to a circle. 
If any point of this circle lie in the periphery of the ellipse, 
(when, intersection being impossible, contact must take place), 
we have, equating the values of SY in the circle and the ellipse, 
(a - rf sin 2 /3 = a 2 (e 2 - cos 2 /3). 
2a — r