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A Treatise on Plane Co-Ordinate Geometry Volume 1; Or, the Application of the Method of Co-Ordinates to the Solution of Problems in Plane Geometry

A Treatise on Plane Co-Ordinate Geometry Volume 1; Or, the Application of the Method of Co-Ordinates to the Solution of Problems in Plane Geometry


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1844 Excerpt: ...(6) represents either two parallel lines or no locus. Hence it appears that the general equation of the second degree wanting the term 2Bxy represents an ellipse, a parabola, a hyperbola, two right lines, one right line, or an isolated point; except in certain cases where it does not represent any locus. The following proposition will complete the discussion of the general equation. Prop. XLI. 187. To shew that the term 2Bxy may be made to disappear from the general equation of the second degree by turning the axes of co-ordinates through a certain angle. If, in the general equation Ax" + 2Bxy 4 Oy + 2Dx + 2Ey + F = 0..'.. (1), we write x cos 0-y sin 0 in place of x, and x sin 0 + y cos 0 in place of y, we turn the axes through the angle 0, leaving the origin unaltered (see Art. 104). When these substitutions for x and y are made, (1) may evidently be expanded and arranged so as to assume the form A'x2 + iSxy + Cy + 2Dx + 2E'y + F = 0 (2), A', B', C, D', &c. being quantities formed from A, B, C, D, E, and 0, in arranging the equation in the form (2). 2Bxy is the only term of (2) we have occasion to know for our present purpose; it can only arise from the three first terms of (1), which, after the substitutions for x and y, become A (x cos 0-y sin 0f + 2B (x cos 0-y sin 0) (x sin 0 + y cos 0) + C (x sin 0 + y cos 0J, or A'x2+2-A cos 0sin 0+2?(cos20-sin'0)+C sinflcos 0xy+C'yi. Hence 2B' = 2B cos 20-(A-C) sin 20. Now, from this expression, it appears that we may make B = 0, by putting 2B tan 20 =---(3), and this we may always do, since the tangent of an angle may be of any magnitude, finite or infinite, positive or negative. Hence it appears that, when 0 is determined from (3), the term 2Bxy does not occur in (2). Consequently, by turning the axes of ...


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Product Details
  • ISBN-13: 9781130713275
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 44
  • Spine Width: 2 mm
  • Width: 189 mm
  • ISBN-10: 113071327X
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 95 gr

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A Treatise on Plane Co-Ordinate Geometry Volume 1; Or, the Application of the Method of Co-Ordinates to the Solution of Problems in Plane Geometry
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