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

250

ANALYTIC GEOMETRY

4. 2x2+ 3xy-2y2- 11x-2 y +12= 0.
Ans. The lines 2x-y-3=0, x+2y-4 =0.
5. x2 + xy + y2+3y+ 4 =0. Ans. No locus.
6. 32-2xy -2y2-_5x-2y-56=0.
7. 2x2-xy+-y2- 7yq+10=0.
8. 4x2-3xy+9y2+ 17'x-12y +19=0.
9. 10x2- 9y - 9y2 +14x +21y -12 = 0.
10. 4x2- 2xy -y2- 4x +y + 5 =0.
11. 22- 3xy+ y2- 6 +5y +4 =-0.
12. Prove that the general equation is of hyperbolic type,
if AC< 0, i.e. if A and C are of opposite signs.
13. The same, if B = 0 and AC = 0.
5. The General Equation, B2 - 4AC = 0. First Method. If
B2 - 4 AC has the value 0, the equation
(1) Ax2 + Bxy + Cy2 + Dx + Ey + F- 0
is said to be of 'parabolic type. The method used in the case
B2 - 4AC( # 0, which begins with shifting the origin so that
the linear terms in x and y drop out, is inapplicable here, since
equations (4) of ~ 4, for the determination of the new origin,
have in general no solution if B2 - 4AC =0.
Let us begin, not with a change of origin, but with a rotation of axes, assuming that B s= 0. Applying to (1) the transformation (2) of ~ 2, we obtain
(2) ax'2 bx'y' + cy'2 + dx' + ey' + F= 0,
where a, b, c are as given by formulas (3) of ~ 3, and
d d= D cosy+Esiny,
(3)
e=- Dsiny+Ecosy.
* They have no solution if the lines
2Ax + By — D — 0, Bx + 2 Cy + E-= 0
are parallel; infinitely many solutions, if these lines are identical.