CONIC SECTIONS, GEOMETRICAL. 89 



axis ; prove that the sum, or the difference, of the focal distances 

 of the feet of these perpendiculars is equal to half the latus 



rectum. 



420. Two points are taken on a parabola such that the sum 

 of the parts of the normals intercepted between the points and the 

 axis is equal to the part of the axis intercepted between the nor- 

 mals; prove that the difference of the normals is equal to the 

 latus rectum. 



421. >ST is the perpendicular from the focus of a parabola on 

 any tangent, a straight line is drawn through Y parallel to the 

 axis to meet in Q a straight line through S at right angles to SY; 

 prove that the locus of Q is a parabola. 



422. At one extremity pf a given finite straight line is drawn 

 any circle touching the line, and from the other extremity is 

 drawn a tangent to the circle ; prove that the point of intersec- 

 tion of this tangent with the tangent parallel to the given straight 

 line lies on a fixed parabola. 



423. Two parabolas have a common focus; from any point 

 on their common tangent are drawn the other tangents to the 

 two ; prove that the distances of these from the common focus 

 are in a constant ratio. 



424. Two tangents are drawn to a parabola making equal 

 angles with a given straight line ; prove that their point of intrr- 

 .scctioii lies on a fixed straight lino passing through the focus. 



425. Two parabolas have their axes parallel and two parallel 

 tangents are drawn to them ; prove that the straight lino joining 



< >ints of contact passes through a fixed point. 



42G. Two parabolas have a common focus S, parallel tangents 

 drawn to them at P, Q meet their common tangent in F t Q \ 

 prove <J is equal to the angle between the 



of the parabolas, and the angle P'SQ' supplementary. 



