Plane and solid analytic geometry; an elementary textbook . 107. The epicycloid.—The path described by a point. Fig. 106. 204 ANALYTIC GEOMETRY [Ch. XV, § 108 on the circumference of a circle which rolls on the out-side of a fixed circle is called an epicycloid. Let the student show that the equations of the epi-cycloid are a + bs x — (a--b) cos (j> — b cosy = (« + b) sin — b sin b a + b * 108. The cardioid. — An important special case of theepicycloid is the cardioid, in which a = b; but instead of

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RM2AJGJ80–Plane and solid analytic geometry; an elementary textbook . 160 ANALYTIC GEOMETRY [Ch. XII, § 86 intersection of tangents at the extremities of chordsthrough any fixed point. This property enables us to construct the polar of anypoint; for any number of points on the polar may be. Fig. 87. determined by finding the intersections of tangents atthe extremities of chords through the point. 2. The polar of any point P1 with respect to a centralconic is parallel to the tangent at the point where the diameterthrough P1 cuts the conic. Y

RM2AJGEJE–Plane and solid analytic geometry; an elementary textbook . a-ticians were the Trisection of an Angle and the Duplication of the Cubeby the aid of ruler and compass alone. It has lately been shown thatthe solution of these problems in this way is impossible. Both problemsinvolve the solution of a cubic equation, and both may be made to dependupon the construction of two mean proportionals between two straightlines. This has been accomplished in various ways by aid of higherplane curves, and it was for this purpose that both the Conchoid andCissoid were invented. 200 ANALYTIC GEOMETRY [Ch. XV,

RM2AJGPG8–Plane and solid analytic geometry; an elementary textbook . pendicu-lar to the tangent through the point of contact. For example, the tangent to the ellipse has been foundto be b2xxx + a2yxy = a2b2. A perpendicular to this linewill have the form a2yxx — b2xYy = k. Since the normalpasses through Pv k = cfiy^^ — b2x^jv and the equationof the normal becomes a2yxx - b2xxy = (a2 - b2)x1yv In like manner, the equation of the normal to thehyperbola is ^ + ^ = ^ + ^ and to the parabola is yxx + my = x1yl + myv The student should note that these formulas apply onlywhen the equations of the curves are i

RM2AJGNYA–Plane and solid analytic geometry; an elementary textbook . Fig. 72. Since the direction of a curve at any point is along thetangent at that point, two curves intersect orthogonally,if their tangents at the point of intersection are perpen-dicular to each other. This is evidently the case here,since the tangent to the ellipse bisects the exterior angle 136 ANALYTIC GEOMETRY [Ch. X, § 78 between the focal radii, and the tangent to the hvperbolabisects the interior angle. The curves therefore intersectorthogonally. 4. The tangent at any point of a parabola makes equalangles with the focal radius

RM2AJH48M–Plane and solid analytic geometry; an elementary textbook . then a simple means of representingposition in a plane by algebraic symbols. This system iscalled the rectangular, and is a particular case of Cartesiancoordinates. In the general Cartesian system the axesare not necessarily perpendicular to each other. In casethey are not perpendicular, the system is called oblique.All the definitions given above hold for the obliquesystem. In Fig. 8, NP is the abscissa of P and MP is its ordi-nate. While rectangular coordinates are more often usedbecause their formulas aresimpler, yet it will occa-s

RM2AJGM6H–Plane and solid analytic geometry; an elementary textbook . related to each other that eachbisects all chords parallel to the other. Such diametersare said to be conjugate to eacli other. Their equations are y = lxx, and y = l2x, where Zx and l2 are connected by the relations givenabove. In the ellipse, lx and l2 have opposite signs, andthe diameters must pass through different quadrants.But in the hyperbola, lx and l2 have the same sign, andthe diameters must pass through the same quadrant. Ineither case, as lx decreases in numerical value, l2 increases,and as one diameter approaches the majo

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RM2AJH23X–Plane and solid analytic geometry; an elementary textbook . he point (— 5, — 5) to the point of division. 7. The coordinates of the vertices of a triangle are (2, 1),(3, —2), and (—4, — 1). Find the equation of the medians,and show that the coordinates of the point of intersection ofany two medians satisfy the equation of the third, and thatthe three medians therefore meet in a point. 8. What are the equations of the diagonals of the rectanglewhose vertices are (0, 0), (a, 0), (0, 6), and (a, b) ? Find thepoint of intersection, and show that they bisect each other. 9. What system of lines is r

RM2AJGGJ5–Plane and solid analytic geometry; an elementary textbook . cus on tangents in the (a) ellipse, (5) hyperbola,(c) parabola. 6. Find the locus of the intersection of tangents at theends of conjugate diameters of an ellipse. Note. — Solve this as a special case of problem 3. 7. Find the locus of the intersection of tangents atthe ends of conjugate diameters of an hyperbola. 8. Radii vectores are drawn at right angles from thecentre of an ellipse. Find the locus of the intersection oftangents at their extremities. 9. Find the locus of the middle point of chords joiningthe ends of conjugate diamet

RM2AJGEA3–Plane and solid analytic geometry; an elementary textbook . value in the second equation and substituting in thefirst, we have x = a vers-1 (-) — V2 ay — y2. But this single equation is not so convenient to use asthe pair of equations from which it was obtained. Thesetwo equations, containing a third variable, are equivalentto the single equation from which 6 has been eliminated.The locus consists of an infinite number of branches,similar to the one shown in Fig. 103, extending both tothe right and to the left of the origin. 106. The hypocycloid.—The path described by a point onthe circumferen

$Plane and solid analytic geometry; an elementary textbook . semicubical parabola, and has the form shown in Fig. 98. 105 19G ANALYTIC GEOMETRY [Ch. XV, § 102 All parabolas are similar to one of these three formsaccording to the value given to n. Let the student plot the locusfor various values of n. Let himalso show that, if n is an eveninteger, or a fraction with an evennumerator and an odd denomina-tor, the curve is similar to theordinary parabola; if n is an oddinteger, or a fraction with an oddnumerator and an odd denomina-tor, the curve is similar to Fig. 97 ;while, if n is a fraction wit Stock Photo$

RM2AJGFT8–Plane and solid analytic geometry; an elementary textbook . semicubical parabola, and has the form shown in Fig. 98. 105 19G ANALYTIC GEOMETRY [Ch. XV, § 102 All parabolas are similar to one of these three formsaccording to the value given to n. Let the student plot the locusfor various values of n. Let himalso show that, if n is an eveninteger, or a fraction with an evennumerator and an odd denomina-tor, the curve is similar to theordinary parabola; if n is an oddinteger, or a fraction with an oddnumerator and an odd denomina-tor, the curve is similar to Fig. 97 ;while, if n is a fraction wit

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