ELEMENTARY PROBLEMS. RECTANGULAR AXES. 
3 
b 2 
Since the ratio of A to ¡u, is not determinate, it follows 
that there is an infinite number of such straight lines, all 
passing through a point of which the coordinates are 
Puissant: Recueil de diverses propositions de Géométrie, p. 192, 
troisième édition. 
4. To determine the position of a point in a given straight 
line, such that the difference between the squares of its distances 
from two given points may be equal to a given area. 
Let the origin of coordinates coincide with one of the given 
points, and let the axis of x contain the other. 
Let a he the abscissa of the latter point, and let the equation 
to the given line be 
x cos a + y sin a = 8 
(1). 
The coordinates of the required point being (¿c, y), we have, 
by the condition of the problem, c l denoting the given area, 
^ + f = d +(x- af + y\ 
The ordinate y is known from (1) and (2), the intersection 
of the two lines, which they represent, being the required point. 
Cor. Suppose the lines represented by (1) and (2) to be 
coincident: then 
ms n On 
whence 
and the value of y cannot be ascertained. 
If then we restrict a and 8 respectively to the values 0, 
of _|_ (f 
and ———, the problem becomes indeterminate, every point 
in the given straight line satisfying the condition of the problem.