. Algebraic geometry; a new treatise on analytical conic sections . na(l +mcos o))=wisinciim sin CO and tan a = 1 + TO cos ojto sin fc) , t lit/ Olll U/ In the same way, tan a =- ; : •^ 1 + m cos 0) .. tan = • TO sm (0 TO sin ft) l+m cos (1) 1 + to cos 0) 1 + / TO sin ft) \ / to sin to \Vl + TO cos ft)/ \1 + m cos ft)/ (m-m) sin ft) , . ,.„ , = 1—r 7^ ;i when simplified. 1 + (m + m) cos 0) + mm ^ Corollary 1. The lines are parallel if to - m = 0. Corollary 2. The lines are at right angles if 1 + (m + m) cos ft) + mm = 0. ART. 71.] THE STRAIGHT LINE. 65 70. If (a^i,. ^i) is a given point on
. Algebraic geometry; a new treatise on analytical conic sections . na(l +mcos o))=wisinciim sin CO and tan a = 1 + TO cos ojto sin fc) , t lit/ Olll U/ In the same way, tan a =- ; : •^ 1 + m cos 0) .. tan = • TO sm (0 TO sin ft) l+m cos (1) 1 + to cos 0) 1 + / TO sin ft) \ / to sin to \Vl + TO cos ft)/ \1 + m cos ft)/ (m-m) sin ft) , . ,.„ , = 1—r 7^ ;i when simplified. 1 + (m + m) cos 0) + mm ^ Corollary 1. The lines are parallel if to - m = 0. Corollary 2. The lines are at right angles if 1 + (m + m) cos ft) + mm = 0. ART. 71.] THE STRAIGHT LINE. 65 70. If (a^i,. ^i) is a given point on a straight line, (x, y) amy pointon the line, a^and ^ the angles the line makes with the axes of x cmd yrespectively, and r the distance between the points (x, y), (aij, ^j), theequation of the straight line may he written x-x^_y-y^I ~ m ~^ , sin/3 sin(w-a) , sin a(= -:—— = —^ -, ana m = -. • sin 01 sm (u Let P be the point (x, y), Q, the point {x^, y^, so that PQ = the ordinates QM, PN, and draw QK parallel to Ox tomeet PN at In the A PKQ, -^ QK PK PQ sin /8 sin a sin to sin/? sin a sinco or ~ ^ = -—^ = r, the read, equations.{ m 11 71. IfXis any constant ax + by + c + X (ax + by+c) = 0 representsa straight line passing through the intersection of the straight linesax + by + c = 0 and ax + by + c = 0. The proof of Art. 34 holds good for oblique axes. E 66 OBLIQUE AXES. [chap. iv. 72. If in any equation we write x + h for x and y + k far y, theresulting equation is the equation of the same locus refewed to parallelaxes through the point (h, k). The proof of Art. 36 holds good. 73. As the point (x^, y.^) moves from one side of the straight linekK+By + O = 0 to the other, the expression ArCj + Byj + C changes itssign. The proof of Art. 37 holds good. 74. To find the length of the perpendicular drawn from the point(ajj, y-i) to the straight line Aa; + By + C = 0, when m is the angle betweenthe axes. Let the straight line cut the axes at E and F
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