292 
HYPERBOLA. 
the centre, and OA\ OB\ two conjugate diameters parallel to 
Cx, Cy, respectively, such that OA : OB':: PM: PE, PE being 
at right angles to Cx. 
To prove that, if the ellipse intersects the two branches in 
P t , P 2 , P 3 , then, P X M X , P 2 if 2 , P,J4 bein g parallel to Cy, ' 
m + W 
De la Hire: Sectiones Conicce, lib. Y. prop. 34. 
Section XII. 
Referred to any Rectangular Axes. 
1. To prove that, in the equation to a rectangular hyperbola, 
referred to any rectangular axes whatever, the coefficients of x 2 
and f are equal with opposite signs. 
The equation to a rectangular hyperbola referred to its 
principal axes is x * — y 2 = a 2 . 
Turning the axes through an angle #, we shall change this 
equation into 
(x cos 6 — y sin 6Y — (îc sin 6 + y cos 6) 2 = a 2 , 
or (îc 2 — y 2 ) cos 20 — 2xy sin 20 = a*. 
If we transfer the origin to any new place, the coefficients 
of x 2 and y 2 will not be affected. The proposition is therefore 
established.