Plane and solid analytic geometry; an elementary textbook . dicular to each other, k = — 2. Substituting thisvalue in the equation and transposing, it becomes (x-y-2y = 4:(x + y-3). Dividing both members by a2 + c2, and both dividing andmultiplying the second member by VDn -f E2, it becomes (^)=2v2(^) V2{X + y-°\ or y2=2V2x, where y is the perpendicular distance of any point (#, y) of the locus from the line x — y — 2 = 0, and where x is the distance from x-\-y—S=0. It is therefore the equation of the locus referred to these lines as X and Y- axes. Construct the two the original equ
Plane and solid analytic geometry; an elementary textbook . dicular to each other, k = — 2. Substituting thisvalue in the equation and transposing, it becomes (x-y-2y = 4:(x + y-3). Dividing both members by a2 + c2, and both dividing andmultiplying the second member by VDn -f E2, it becomes (^)=2v2(^) V2{X + y-°\ or y2=2V2x, where y is the perpendicular distance of any point (#, y) of the locus from the line x — y — 2 = 0, and where x is the distance from x-\-y—S=0. It is therefore the equation of the locus referred to these lines as X and Y- axes. Construct the two the original equationwe see that the curvetouches the X-axis atthe point (4, 0), and doesnot cut the F-axis. It is then easily seen which FlG- 91- is the positive direction of the axis OX, and the curve can beplotted as in Fig. 91. 2. Plot by this method the locus of the following equations:(a) x2 - 2 xy + y2 - 6 x - 6 y + 9 = 0,(6) x2 + 6 xy -f- 9 y2 + x - 6 y - 9 = 0, (c) 2x2 + $y2 + 8xy + x + y + 3 = 0, (d) f - 2 x - 8 y + 10 = 0, (e) 4:X2 + 4xy + y2 + 6 = 183 ANALYTIC GEOMETRY [Ch. XIII, § 97 when A = 0, • 97. Summary. — It has been shown in this chapter thatthe general equation of the second degree represents, and when B2 — 4 AC 0, an hyperbola; and when B2 — 4 AC 0, two real intersect-ing lines. All of these forms may be obtained as plane sections ofa right circular cone, and are all included under the termconies. An equation of the second degree must thereforerepresent some conic either in its regular or degenerate form. PROBLEMS Determine the nature of the locus of each of the followingequations: 1. 3x?-2xy + y2 + 2x + 2y + 5 = 0. 2. x2 + xy + y2 + 2 x + 3y -3 = 0. 3. 2X2- 5xy- 3/4-9 x- 13 y + 10 = 0. 4. 4z2 + 2xy-\f + 6x + 2y + 3 = 0. 5. 9x2- 12xy + ±tf-
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