The equation to a tangent at the point (~11) is x Y1+x1y = 1⁄2 (a2 + b2). The area of a triangle contained between these axes and a tangent is constant, and = (a + b2) sin x,y. If UWu is parallel to Ci, then UW = Wu. (H. C. S. 245—59; Biot, 250—8; Hust. Prop. 11—5.) DISCUSSION OF LINES OF THE SECOND ORDER. (21.) The general form of an equation of the second degree containing two unknown quantities is ay2+ bxy+cx2 + dy+ex+f=0. From the solution of this equation (b2 — 4ac) x2 + 2 (bd — 2 ae) x + ď2 — 4aƒ3 ; (b2 — 4ac) y2 + 2(be — 2cd) y + e2 — 4cƒ |3 . which values of x and y may be thus represented: A straight line cannot cut a line of the second order in more than two points. The locus of the middle points of any number of parallel chords is a diameter. If the chords drawn parallel to any diameter are bisected by another diameter, then the chords parallel to the latter, will be bisected by the former. The co-ordinates of the centre of the curve are If b=0, a and c must have the same sign in the ellipse, and different signs in the hyperbola. If the curve is an ellipse or hyperbola, and the axes parallel to any system of conjugate diameters, the form of the equation ay2+cx2 + dy+ex+f=0. is If the origin is at the centre of the curve, the equation becomes ay2+bxy+cx2 +f=0: and if the curve is also referred to conjugate diameters, then ay2+cx2 +ƒ=0. If the axes are parallel to the asymptotes of an hyperbola, the form of the equation is bxy+dy + ex+f=0. If e2>4cf, the curve intersects the axis of x in two points; touches the axis of x; e=4cf,.. does not meet the axis of x; The same conditions exist with regard to the axis of y, according as d2 >, =, or <4af. If B2>AC, the curve intersects the diameter, y=— bx+d ; 2 a (22.) Determination of the species, and their varieties. are equal, the ellipse is reduced to a point. line. If the roots are impossible, the ellipse becomes an imaginary If a=c, and b=0, the ellipse becomes a circle. [2] If A>0, the curve is an hyperbola. If the roots of the equation (a) are possible, the diameter, cuts the curve; if impossible, that diameter does not meet it. If the roots are equal, the hyperbola becomes two intersecting straight lines. If a=c, and b=0, the hyperbola is equilateral. [3] If A=0, the curve is a parabola. If B also =0, and C>0, the equation represents two parallel straight lines, (H. A. G. 81-89, 184-213; G. G. A. Ch. iv; Biot, 264-304.) SUMMARY OF EQUATIONS. (23.) Rectilinear equations. [1] Let the curve be referred to its centre, and principal . . . . equilateral hyperbola, y = x2 — a2. [2] Let the curve be referred to the principal diameter, and a tangent at the vertex, then = 2 a2 (2 a x − x2); (2 a x + x2); y2 = 4mx; y2 = 2ax-x2; equilateral hyperbola, y2 = 2ax + x2. The general form of the equation is y = mx + nx2. In the Indee The same equations subsist, if a and b are any system of conjugate diameters to which the curve is referred. (24.) Polar equations. [1] Let the curve be referred to the radius vector, and angle contained between it, and the principal diameter, then [2] Equations between the radius vector, and a perpendicular on the tangent from the pole. ANALYTICAL GEOMETRY OF THREE DIMENSIONS. (25.) The straight line. The equations to a straight line referred to three rectangular co-ordinates, are which are the equations to its projections on the planes of xx, yx respectively. The equations to a line passing through the point (x1,y1,1), are The equations to a line passing through two points (~1,Y1,≈1), (x2,y2,2), are The co-ordinates of the point of intersection are The distance (d1) of a point (≈111) from the origin; 2 2 12 d1 = (x2+y12 +≈ ̧3)ŝ=≈(1 + a2 +b2)§ ; if x=ax, y=bs are the equations to the line passing through the given point, and the origin. The distance (D) between two points (x1,y1,1), (X2,Y29*2) ; D= (x1− x ̧)2 + (Y1 − Y2)2 + (≈1 − ≈2 · 2 ( x1 x2 + Y 1 Y 1⁄2 + ≈182) do being the distance of (~,,2,*) from the origin. |