108 BOOK OF MATHEMATICAL PROBLEMS. 



II. Parabola referred to its axis. 



The equation of the parabola being taken y 2 4o#, the co- 

 ordinates of any point on it may be represented by f ,, --J, 



and with this notation, the equation of the tangent is y = mx + ; 



of the normal my + x = 2a + s ; and of the chord through two 



points (wij, W 2 ), 2m ^ x y (n^ + w 2 ) + 2a = 0. The equation of 

 the two tangents drawn through a point (X, Y) is 



As an example, we may take the following, " To find the locus 

 of the point of intersection of normals to a parabola at right 

 angles to each other." 



If (X, Y) be a point on the locus, the points on the parabola 

 to which the normals are drawn from (X, Y) are given by the 

 equation 



and if m lt w a , m 3 be the three roots of this equation, 



_2a X _-. _ a 



also since two normals meet at right angles in the point, the pro- 

 duct of two of the roots is 1. Let then m a m a = 1. Then 



_-a _3a~X_~:r > 



or the locus is the parabola 



7 2 = a(X-3a). 



Again, " The sides of a triangle touch a parabola, and two of 

 its angular points lie on another parabola having the same axis, 

 to find the locus of the third angular point." 



