110 BOOK OF MATHEMATICAL PROBLEMS:. 



551. TP, TQ are two tangents to a parabola ; prove that the 

 perpendiculars let fall from P, T, Q on any other tangent are in 

 geometric progression. 



552. Four fixed tangents are drawn to a parabola, and from 

 the angular points taken in order of a quadrangle formed by them 

 are let fall perpendiculars p lt p a , p a , p^ on any other tangent ; 

 prove that p l P a = P a P 4 - 



553. The distance of the middle point of any one of the three 

 diagonals of a quadrilateral from the axis of the parabola which 

 touches the sides is one fourth of the sum of the distances of the 

 four points of contact from the axis. 



554. Through the point T, where the tangent to a given 

 parabola at P meets the axis, is drawn a straight line TQQ' 

 meeting the parabola in Q, Q', and dividing the ordinate of P 

 in a given ratio ; prove that PQ, PQ' will both touch a fixed 

 parabola having the same vertex and axis as the given one. 



555. Two equal parabolas have their axes in the same 

 straight line, and from any point on the outer tangents are drawn 

 to the inner ; prove that these tangents will intercept a constant 

 length on the tangent at the vertex of the inner. 



556. If p, q, r be the perpendiculars from the angular points 

 of a triangle ABC, whose sides touch a parabola, on the directrix, 

 and x, y, z perpendiculars from the same points on any other 

 tangent, 



p tan A + q tan B + r tan (7=0, 



p tan A q tan B r tan C A 

 and - + - -+- - = 0. 



x y z 



557. A tangent is drawn to the circle of curvature at the 

 vertex of a parabola, and the ordinates of the points where it 



meets the parabola are y^ y a ; prove that = , 2c being 



y \ MS 



the latus rectum. 



