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

178

ANALYTIC GEOMETRY

Substituting these values of / in turn in (7), we obtain
4x- y+5 -=0, 4x-y-5=0,
as the equations of the two tangents of slope 4 to the ellipse
(6). From equations (9) and (10) it follows that the points of
contact of these tangents are, respectively, (- 1, 1) and
(i, - i).
Both the methods described in this paragraph are general
in application. For the usual type of problem met with in a
first course in Analytic Geometry either method may be used
with facility. It is, however, to be noted that the second
method presupposes that the equation of the tangent to the
curve at an arbitrary point on the curve is known, whereas
the first does not. Accordingly, in case a curve is given, for
which the general equation of the tangent is not known, -for
example, the parabola, y = 3x2- 2x + 1, -the first method
will be shorter to apply.
EXERCISES
Determine in each of the following cases how many tangents there are to the given conic with the given slope. Find
the equations of the tangents and the coordinates of the points
of tangency. Use both methods in Exs. 1, 2, 3, checking the
results of one by those of the other.
Conic Slope
1. x+y2 =5, 2.
Ans 2x -y- 5= O, tangent at (2, - 1),
2x - y + 5 = O, tangent at (-2, 1).
2. y2 =3x, 4
3. 2x+=2 12,
4. x2+8y=0, 2.
5. 42 - y2 =20, 3.
6. 2 + y+2 =0, '
7. 6y2-5x=0, 13.