CONIC SECTIONS, ANALYTICAL. 



C40. Two points (a?,, y,), (x a , y a ) are taken on tlic 

 = c*', prove that the equation of the line joining them 



-5- + -*-=!. 



t/ 



647. Tlie equation of the chord of the hyperbola _ ?L = 



fl 6 

 which is bisected at the point (X, Y) is 



648. The locus of points whose polars with respect to a 

 given parabola touch the circle of curvature at the vertex is a 

 rectangular hyperbola. 



G40. A double ordinate PP' is drawn to the ellipse - t + = 1, 



a 6 



x 9 v* 

 lie tangent at P meets the hyperbola -; -?? = 1 in Q, (?', 



prove that F'Q, PQ are tangents to the hyperbola ; and, if R t K 

 be the points in which these lines again meet the ellipse, that 

 livi.l, -s V'/' into two parts in the ratio of 2 : 1. 



lc is drawn to touch the asymptotes of a hyper- 

 bola ; prove that the tangents drawn to it at the points where 

 it meets ti ola will also touch the auxiliary circle of the 



hyperbola, 



I . Two hyperbolas have the same asymptotes, and XPQ is 



:> parallel to one asymptote meeting the ot)i< r in .V, and tho 



/>,<?; a tangent at Q meets the outer hyperbola in two 



% and the lines joining these to the centre meet the ordinate 



o L, M prove that NQ is a geometric mean between XL, 



<1 tint \ / is a harmonic mean between NQ and the liar- 



io mean between NL 9 NM. 



i ho locus of a point from which can be drawn two 

 -,'ht lines at right angles to each other, each of which 



