70 Journal of the Mitchell Society [December 



four corners of the niaximuiii rectangle inscribed the ellipse satisfy 



the equation of the rectangular hyperbola xy = ^—, Accordingly 



the ellipse and the rectangular hyperbola, having two points in com- 

 mon, must either intersect each other or be tangential at the given 

 points. 



The tangent to the ellipse at the point (x,, y,) has the form 



+ a 

 and hence we have for the tangent to the ellipse at the point .— , 



V 2 



V2' 



ax by 



aVT ^ bVT" ^ 

 or 



X y , 



- + Y-=V2 ■■••(3) 



The tangent to the rectangular hyperbola at the point (x,, yO 

 has the form 



yix + xi y = ab; 



and hence Ave have for the tangent to the rectangular hyperbola at 

 + a + b 



the point 



that is 



V2' V2 



bx ay 



+ ~~r^ = ab, 



\/2 VT" 



— + ^ = V 2 ... (4) 

 a b 



As equations (3) and (4) are identical, it is clear that the ellipse and 



the rectangular hyperbola have a common tangent at the point 



+ a + b 

 ,-^ , ,— It may be noted that, since the rectangular hyperbola 



consists of two branches lying respectively in the first and third 

 quadrants, the ellipse and the rectangular hyperbola cannot intersect 



