Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

158

ANALYTIC GEOMETRY

We have, then, as the final result: The slope of the curve (6),
at an arbitary point (xi, Yi) on it, is
a2
-^
x12
Equation of the Tangent. Since the tangent to the curve (1),
y = x2,
at the point (1, 1) has the slope 2, its equation is
y-1 = 2(x- 1), or 2x-y —1 = 0.
Similarly, the equation of the tangent to the curve (1) at an
arbitrary point P: (xi, yj) is
Y - = 2 x(x - xi),
or
Y -Y1 = 2 xx -2 x12.
This equation may be simplified by use of the equality,
Yi = 12,
which says that the point P lies on the curve. For, if we replace the term 2 x2 by its equal, 2 y, and then combine the
terms in y1, the equation becomes
y + yi = 2 xjx.
This equation of the tangent is of the first degree in x and y,
as it should be. The quantities x1 and yi are the arbitrary,
but in any given case fixed, coordinates of P and are not
variables.
Equation of the Normal. The line through a point P of a
curve perpendicular to the tangent at P is known as the
normal to the curve at P.
Since the tangent to the curve y = x2 at the point (1, 1)
has the slope 2, the normal at this point has the slope -2.
Consequently, the equation of the normal is

y- 1 = - (x- 1),

or x+2y-3=O.