An Elementary Treatise on Analytical Geometry, with Numerous by W J. Johnston

By W J. Johnston

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Given in one of these forms is we may reduce others. x a v b Ex. Express - + £ = Here 1 in the y r- b = Thus form y x a l m = mx + y J , = a , b c. = a c = u x+b. b. it : The 59-] § 59. To reduce Straight Line Ax + By + C = + y sin 06 — p = o. ABC • 2 + B since the 2 sum unity — +B a C be negative we COS 06 =— A >/A + B 2 = " " C 2 write } +B +B 2 P If VA and 2 if ; form then ; y + • 2 2 the of the squares of /A is -7^== 7A + B x + BT >/A 2 + Divide by Now oto the equation x cos 06 VA 39 C VA +B 2 2 then changing the signs of all the terms the be written -Ax-By-C = o; the preceding method is then applicable.

J Straight Line 33 Then Ax + By + C = 0, x x 2 Ax + By A, B, C from 3 3 We may Ax + By + C = o. eliminate -x )+ 1 2 •'• 3) 3) and the second by yx - y2 then divide by A. ; -x) (*! (y2 2 - - y3) zero ; -x (x 2 3) -y = (yx = 2) o o Xi yx i x x3 y2 1 y3 1 whose vertices are the area of the triangle is third -y = o - y = o. e. (x 3 y3 ) B(yi A (x - x + B (y of these by y — y 2 and subtract o, This gives A(x Multiply the + C = these equations. Thus, subtract the second equation from the from the second.

06 p = o. Alternative proof. The equation x v ~= = UA + OB = ON = OA cos OC to the line But p p = ON = OB NOB = OB cos sin Substitute these values of x _ A f_p_) \cos a/ OA y is OB and = a a then the equation becomes : x or I, sin cos a + y sin (X = p. ) \sm a/ § 56. The form x cos straight line (§52). i -=r-r- JOA= cosCX ,. Also is called the ' 06 + y sin at = standard ' p of the equation to a form. be assumed as a convention that p is always positive, and also that 06 is always positive, i.

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