. Algebraic geometry; a new treatise on analytical conic sections . onjugate diameters of the original hyperbola. 278. If a hyperbola has its centre at the origin, its equation containsno terms of the first Art. 213. *279. To find the equation of a hyperbola referred to two conjugatediameters as axes of co-ordinates. (Oblique.) The origin being at the centre of the curve, if (x, y) lies on thecurve, {-X, -,y) also lies on the curve. .. its equation can contain no terms of the first degree. We may therefore take Aa;2 + 2Ha;y+B2/2=l (1) to be the equation of the hyperbola. Again if P
. Algebraic geometry; a new treatise on analytical conic sections . onjugate diameters of the original hyperbola. 278. If a hyperbola has its centre at the origin, its equation containsno terms of the first Art. 213. *279. To find the equation of a hyperbola referred to two conjugatediameters as axes of co-ordinates. (Oblique.) The origin being at the centre of the curve, if (x, y) lies on thecurve, {-X, -,y) also lies on the curve. .. its equation can contain no terms of the first degree. We may therefore take Aa;2 + 2Ha;y+B2/2=l (1) to be the equation of the hyperbola. Again if P is the point {x,y) and the chord PVP is drawnparallel to CB the axis of y, PV = PV, or the co-ordinates of P, apoint on the curve, are {x, - y). K 258 THE HYPERBOLA. [chap. XII. .. Aa;2 _ 2HX1/+ By^=l, by substitution. ., from (1) by subtraction, H = 0. .. the equation of the curve reduces to Aa;2 + By2= the curve cut the axis of x at A and let CA = a. When y = 0, x = a. :. Aa2=l, and A = axis of y does not meet the curve; .. B must be FiQ. 167. Taking it equal to - t^, the equation of the curve is The point (asec 9, btan 6) lies on the curve for allvalues of d. Examples XII. e. 1. Find the pole of the chord 21a;-9«=28 with respect to the hyper-bola Ta: - 12j^2= 112. 2. In the hyperbola 25a;-16^=400 find the equation of the diameterconjugate to 3y = x. 3. Find the condition that y + = 1 may be a tangent to the hyperbola O TO *■* Exs. XII. THE HYPEEBOLA. 259 4. In the hyperbola ^ - ^2 = 1 find the condition that the chord joining the two points (aseofl, 6 tan 9), (aseo0, 6tan.^) should subtend a Tightangle at the osntre. 5. P is a point on a hyperbola, Q on the conjugate hyperbola • if PCQis a right angle pg^ - p^ is constant. 6. In any hyperbola the circle on SP as diameter touches the auxiliarycircle. 7. Find the equation of the chord of the hyperbola iex^-9i/^=:liiwhich is bisected at the point (12, 3). 8. Find the length of the semi
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