. Algebraic geometry; a new treatise on analytical conic sections . FlQ. 27. Draw the ordinates PN, QM, and draw QK perpendicular to PN. From the A PKQ, QK = PQ cos 9ovx-x,=r cos 6 and ^^ = r. ■ cos 0 Also PK = PQsine, or y-y^ = r& and ^^^^ = r. sin d cos 6^ sin S ART. 32.] THE STRAIGHT LINE. 31 31. The angle between the straight lines y = mx + c, y = mx + d is \l+mmyIf AB, AC are the straightlines, I. PiBx = tan~^TO = 6 suppose,L ACx = tan^m= & if <^ = , theangle required, .. tan ^ = tan (0-0) _ tan 6 - tan 61 + tan 0 tan ffm-m!. 1 +mm .. <fi = tan .i^m-m \yl+mmJ F
. Algebraic geometry; a new treatise on analytical conic sections . FlQ. 27. Draw the ordinates PN, QM, and draw QK perpendicular to PN. From the A PKQ, QK = PQ cos 9ovx-x,=r cos 6 and ^^ = r. ■ cos 0 Also PK = PQsine, or y-y^ = r& and ^^^^ = r. sin d cos 6^ sin S ART. 32.] THE STRAIGHT LINE. 31 31. The angle between the straight lines y = mx + c, y = mx + d is \l+mmyIf AB, AC are the straightlines, I. PiBx = tan~^TO = 6 suppose,L ACx = tan^m= & if <^ = , theangle required, .. tan ^ = tan (0-0) _ tan 6 - tan 61 + tan 0 tan ffm-m!. 1 +mm .. <fi = tan .i^m-m \yl+mmJ FlQ. 28. Corollary 1. If the two straight lines y = mx + c, y = mx + care parallel, m = m. This is evident from a diagram, for the lines make equal angleswith Ox, and therefore their slopes (or gradients) are equal. Corollary 2. If the two straight lines y = mx + c, y = mx + care at right angles, mm = - 1. The tangent of the angle between them =:, ;. ° 1+ mm rn-i rm But tan 90° = 00 : .. :;—■ ; = oo • . mm +1 =0, ormm= - 1. 1 + mm 32. If the two straight lines a^x + b^y + Cj = 0, agX + b^y+Cj = 0are at right angles, ajaj + bjbg = 0. The Ml or slope of the first line = --^. „ second ,, = - -r^ On 32 THE STRAIGHT LINE. [chap. II. .. by the preceding article {-fr)^{~]f)-~^ (*™ = ~ 1) or a-fi2 + ^1^2 — ^•Note carefully the following illustrations : (1) 5x-2y=3, 5x-2y = k are parallel lines for all values of k, for theirslopes are equal. (2) 5a;-2y = 3, 2x + 5y = k are perpendicular lines for all values of k,for a^a^ + b-^b2=5 x 2 -2 x 5
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