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ANALYTICAL GEOMETRY 45 



Now, if we are to find the points of intersection of the 
straight line 2x+2y=l and the circle x 2 + y z =l, we seek 
those points which are common to both curves, that is to say, 
all the pairs of values of x and y which satisfy both the above 
equations. Thus we need not trouble about the geometrical 
picture, but we only have to apply the rules of algebra for 
finding the values of x and y which satisfy two " simultaneous " 
equations in x and y. In the above case, if (X, Y) is a point 

1 2X 

of intersection, we have F = -^ and therefore, by substi- 

a 

f\ - 2Y\ 2 
tution in the other equation, X z + ( - J = 1. This gives 

a quadratic equation 

8X 2 - 4.X -3 = 

for X, and, by rules, we find that X must be either J (1 + >/7) 
or j(l- /s/7). Hence there are two values of the abscissa 
which are given when we ask what are the co-ordinates of the 
points of intersection ; and the value of y which corresponds to 
each of these re's is given by substitution in the equation 



Thus we find again the fact, obvious from a figure, that a 
straight line cuts a circle at two points at most. We can 
determine the points of intersection of any two curves .whose 
equations can be expressed algebraically, but of course the 
process is much more complicated in more general cases. Here 
we will consider an important case of intersection of a straight 
line. 

Think of a straight line cutting a circle at two points. 
Imagine one point fixed and the other point moved up towards 
the first. The intersecting line approaches more and more to 
the position of the tangent to the circle at the first point, and, 
by making the movable point approach the other closely enough, 
the secant will approach the tangent in position as nearly as 
we wish. Now, a tangent to a curve at a certain point was 
defined by the Greeks as a straight line through the point such 
that between it and the curve no other straight line could be 
drawn. Note that other curves might be drawn : thus various 
circles may have the same tangent at a common point on their 
circumference, but no circle and no curve met with in ele- 
mentary mathematics has more than one tangent at a point.