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Extra info for An Elementary Treatise on Analytical Geometry, with Numerous Examples
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. e. measured round in the positive It will direction from § 57. With OX. this understanding, it will be found on examination that whatever be the position of the line the proofs in Arts. 54 and 55 are perfectly general, and that in all cases the equation is x The cos OC + y sin OC = p.
O. or , 2 = CX 1 20 . to illustrate these examples. Exercises 1. Express the following equations x + y= x^3 3, x >/3 + Ans. OC = 45°, p = 3 2. y. mx + m =Vi m + 2 3 = OC ; a =- % 20 330 sin OC T / , = o. p = = if, 3 p a= ; - o, 150°, ff x y - + <-=!. a b c, —==- y — 1 x + Vi + 2 bx Va cos ; = p , : xv 3-y+6 = o, form in the standard y = standard form + 210 3 in the -xV3 y+6 = o, 5 x — 12 y + 6-o, y — 6 - + y + — Va2 + b2 V 1 + nrr == = ab ay + + b2 m r» 2 Va 2 + b o. 2 PROJECTIONS 8 60i proof in § Deff is If the properties of projections are assumed, the statement of the 54 may — If A' be simplified.
Y2 = x2 5. Express in polar co-ord's the eq'n, 6. Express 2 in rect' co-ord's the eq'n, r sin 2 7. What § 4|. loci are represented In fig', area by the equations r x positive or negative according as is The reader will see on reflection that We can Take A X which the points O, P, — now a2 2 = = Ans. r2 cos 2 . Ans. xy . = a2 a2 2 is positive . = —? 5, - ; 1 — r2 sin (0 X 2 is Q are mentioned otherwise it is 2 ). positive or negative. we go round if - ; the triangle then if this OPQ order negative.