390 Cones and conics
32 This long problem should be very easy. The problem is to read about
graphs of equations of the form
(1) 11x2+By2+Cx+Dy+E=0
and to verify correctness of each assertion that is not completely obvious. In
case .4B > 0, completion of squares enables us to putthe equation in the form
(2) -4(x - xo)2 + B(y - yo)2 = K,
where A and B have the same sign, and the graph is an ellipse or a circle or a
single point or the empty set. In case .4B < 0, the equation (1) can be put in
the form (2), where .4 and B have opposite signs, and the graph is either a hyper-
bola (when K 0) or a pair of intersecting lines (when K = 0). In case .4 =
B = 0, the graph is a line or the empty set (in case C = D = 0 and E FA 0) or
the whole plane (in case C = D = E = 0). So far we have covered all situations
except those in which one of -4 and B is zero and the other is not. The case in
which .4 = 0 and B 0 0 being analogous, we suppose henceforth that .4 0
and B = 0 and that (1) has been reduced to
x2 + Cix + D1y + E1 = 0.
In case D1 0 0, completion of a square gives the equation y - yo = K(x - xo)2
and the graph is a parabola. In case D, = 0, the graph consists of two vertical
lines or a single vertical line or the empty set. The results may be summarized.
Depending upon the values of -4, B, C, D, E, the graph of (1) may be a conic
(an ellipse, a hyperbola, a parabola, a circle; two intersecting lines, a single line,
or a single point) and it may be a set which is not a conic (two distinct parallel
lines, a plane, or the empty set). Still more information is available. The
equation (1) is said to have elliptic type when AB > 0 even though there are
cases in which the graphs are circles or points or empty sets. Similarly, (1) is
said to have hyperbolic type when AB < 0 and to have parabolic type when -4
and B are not both 0 but AB = 0.
33 Let B and B be nonzero constants not both negative. The graph of the
equation(1) .4x2+Bye-1
is then a central conic K (a circle or ellipse or hyperbola) having its center at the
origin 0. When Pi(xi,yj) is a point different from the origin, the equation
(2) Axlx + Byiy = 1
is the equation of a line L1 which is called the polar line (with respect to the conic
K) of the point P1. Moreover, the point P1 is called the polar point of the line
L1. These polar points and lines are very important in some parts of mathe-
matics, and we can start learning about them.
Figure 6.498 If the point P1 lies on the conic K, then L1, the
polar line of Pi, is the line tangent to K at P1.
Q1 When P1 is not on K, matters are much more
interesting. Suppose first that P1 lies "out-
L1 P(x1,y1) side" the conic K so that there exist two points
R, Q, and R1 on K such that the tangents to K at
Q, and R1 contain P1. Then, as Figure 6.498