The condition that (x', y') be on (3) is x' cos a + y' sin a = p'. Subtracting p from both members, we find cos a+ y' sin a p, the distance from (x', y') to x cos a + y sin a p =O.
Angle between Let st. be the angle between two lines whose slopes are tri, and m,. Then tan 4 ----- :(1 + mono). If 4-0, then tan 4 °=0, and m2= m2, condition of parallelism. If 4-90°, then tan 4 = co , and 1 + mim,= 0, whence mirtici-- 1, condition of perpendicularity.
The Circle and Its About any point (a, b) as center describe a circle of any radius r. Let P(x, y) be any point of the circle. A glance at the figure shows that x and y are connected by the relation (xa)'+ equation of the circle. By comparison with the equation x' + 2Ax 2By C=0, the latter is found to represent a circle of center (A, B), and radius VA' B' C. If the radicand be negative, the radius and hence the circle is imaginary. The tangent (1) to the circle at the point (x,, yi) is (x, a) (x a) + (yi b) (y b) r' =0; the normal (2) is (3/1 (x1 a) (31 yL) If the center be at the origin, the circle is x' + the tangent at (x,, y,) is x,x + y,y r'°-.0, and the normal is The line y=mx + c is tangent to x' + y' r' = 0 when and only when the distance from 0 to the line is r, i.e., when c.° ± rV 1+ pre; hence y = mx ± eV] + m' is tangent to the circle and as m varies completely envelopes the circle. Similarly it may be found that for every value of m the line yb = m(x a) ±-. rN/ 1+ tangent to the circle of center (a, b) and radius r. This equation (in slope form) of the tan gent is sometimes called the magic equation of the tangent. By transformation of co-ordinates or otherwise the polar equation of the circle of center (p', 8) and radius r may be found to be P' 2P' p coS(e P' The Conic Sections: Ellipse, Parabola, The equation, of the path locus, or curve generated by a point (x, y) which so moves that the ratio of its distance from a fixed point (called the focus) to its distance from a fixed straight line (called the directrix) is a constant e (called the eccentricity) is of the form Bya+ Cxy Dx ± Ey + F =0; conversely, the locus of every equation of sec ond degree is a curve of the kind in question. All such curves are called conics or conic sections, because any one of them is the inter section of a plane and a cone, where by cone , meant the surface that may be generated by a straight line revolving about a fixed point (vertex) and making a constant angle with a fixed line (axis) through the point; conversely, every such intersection is a conic, curve of second degree. The conics, of the very greatest importance alike in pure and in applied mathematics (cf. Asmaitomv), were
studied by the ancients (cf. Aeou.oxius), who conceived then? as intersections of plane and cone. There are three species of conics, a conic being named ellipse (E), parabola (P), or hyperbola (H), according as the eccentricity e C, or > 1. The foregoing equation repre sents an E, a P, or an H, according as 4AB C' is positive, zero, or negative. Among the conics of any species are degenerate or de graded or so-called improper conics of that species. The degraded form of E is any pair of imaginary straight lines intersecting in a real point (center of the conic, vertex of the cone) ; any pair of parallel real straight lines is a degraded P; and any pair of non-parallel real lines is a degenerate H. By suitable trans formations of co-ordinates the equation of any conic may be nude to assume a simplest, named standard, form. The standard forms of the species are presented below.
Ellipse. The standard form of the E is y' in which a is the half major and as b' b the half minor, axis of the E. The focus and directrix are respectively F and DD or F' and D'D'. The equations of the directrices are x =. a :e ; e c :a. The sum of the focal radii of any point of E is p -F pl a prop erty often taken as definition of the E, instead of the relation p ed. If b a, the E is a circle; hence the circle is an E of coincident foci and eccentricity zero. The area of E is nab.
Hyperbola. The standard equation is a' b' 1, where a is half of the transverse,and b is half of the conjugate, axis. The equa tion represents the curve composed of the two branches (1) and (2). For any point, p' p=2a,si defining property of the H. The H composed of (1') and (2') is called conjugate ' y' to the other one and has x -- = 1 for its equation. The two oblique lines through 0 tangent to both H's at oo are named asymptotes of the curves. The equations of the asymp totes are y= ± a x. The corresponding lines of the E are imaginary, i a x. If a =--- b, the H, = a', is called equiaxial or equi lateral or rectangular, its asymptotes being at right angles. It is related to the general H as the circle to the general E.
Parabola. The standard equation I is 32= 4px. The equation of the directrix is x =-- -p the co-ordinates of the focus are (p, 0) for any point p = d. The second foals and the center lie at oo on the axis of the curve.
For some account of plane curves of higher order or degree, see the article HIGHER PLANE CURVES, and for elaborate and detailed treat ment of the analytical geometry of the plant, including exhaustive discussion of the conics, see works cited in bibliography below. We add here a note introducing the Cartesian geometry of