

CONIC SECTIONS, ANALYTICAL. Ill 



558. OA, OB are two tangents to a parabola meeting the 



tangent at the vertex in P, Q \ prove that 



* PQ = OA cos QPA = OB cos PQB. 



.. Two parabolas have a common focus and direction of 

 / a chord of the outer is bisected by the inner in P, 

 /'/' parallel to the axis meets the outer in P'; prove that c 

 a mean proportional between the tangents drawn from P' to the 



560. The locus of the centre of the Nine Points' Circle of a 

 triangle, formed by three tangents to a parabola, of which two are 

 . is a straight line. 



1 . Prove that the parabolas y* = 4oa;, y f + 2cy + lax = 8a f , 

 ach other at right angles in two points. 



562. Through each point of the straight line * + *jf 1 is 



drawn a chord of the parabola y f = 4oa; bisected in the jK>iiit; 

 prove that this chord touches the parabola 



563. Two equal parabolas have a common focus and axes in 

 the same straight line ; from any point of either two tangents are 

 drawn to the other : prove that the centres of two of tin- 



which touch the sides of the triangle formed by the tan- 

 gents and their chord of contact lie on the parabola to which the 

 tangents are drawn. 



564. The two parabolas y* = ox, y* = 4a(a> + a) are so re- 

 lated that if a normal'to the latter meet the former in /'. /", and 

 A be the vertex of the former, AP or AP' is perpendicular to the 



nal. 



665. The normals at three point* of the parabola y* - 4<w 

 meet in the point (h, k) ; prove that tho equation of the . 

 iirce points is 



