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Art.

122. The length of the perpendicular from the centre on the tangent,

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125. If CE is drawn parallel to the tangent, meeting HP in E, then PE AC 126. The equation to the normal

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130. All the diameters of the ellipse pass through the centre;

y = ax + c, a2a y + b2x = 0, are the chord and corresponding diameter 131. There is an infinite number of pairs of conjugate diameters;

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133. Equation to the curve referred to any conjugate diameters,

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136. The sq. on QV: the rectangle P V, VP' :: sq. on CD: sq. on CP
137. The ellipse being referred to conjugate axes, the equation to the tangent is

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138. The ellipse being referred to its axes, the tangent is parallel to the conjugate diameter: the two equations are,

a2 y y' + b2 x x = a b2, the tangent,

a2 y y' + b2 xx′ = 0, the parallel conjugate

139. The square upon CD = the rectangle SP, HP 140. The perpendicular from the centre on the tangent,

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141, 142. The product of the tangents of the angles which a pair of supplemental chords makes with the axis major is constant,

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143. The tangent of the angle between two supplemental chords,

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144. Supplemental chords are parallel to conjugate diameters

145. The equation to the ellipse, referred to its equal conjugate diameters, is

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X

Art.

148. The focus, the pole, r =

a (1-e2)

-e cos.

2a (1-e2) cos.

149. The pole at the vertex, u =

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155-7. Discussion of the equation:

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The sq. on MP: rectangle A M, M A':: sq. on BC: sq. on A C .

158. The equation to the equilateral hyperbola is

y2 — x2 = — a2

159. The results obtained for the ellipse are applicable to the hyperbola, by changing

b2 into - b

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160. The Latus Rectum is defined to be a third proportional to the transverse and conjugate axes

161-3. The focus; the eccentricity:

The rectangle AS, S A' = the square on B C

ex-a, HP ex+a, HP – SPAA'

164. SP

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165. To find the locus of a point the difference of whose distances from two fixed points is constant

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168. The equation to the tangent, at the extremity of the Latus Rectum, is

y = ex- a.

The distances of any point from the focus and from the directrix are in a constant ratio

169. The length of the perpendicular from the focus on the tangent,

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170. The locus of y is the circle on the transverse axis

171. The tangent makes equal angles with the focal distances,

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Art.

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174. If CE be drawn parallel to the tangent and meeting HP in E, then P EAC 98 175-7. The equation to the normal is

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The rectangle P G, PG' the rectangle SP, HP

178, 9. The diameters of the hyperbola pass through the centre, but do not all meet

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180, 1. There is an infinite number of pairs of conjugate diameters,

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182. The equation to the curve referred to conjugate axes is

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185. The sq. on QV: the rectangle P V, V P' :: sq. on CD: sq. on CP 187. The conjugate diameter is parallel to the tangent. The equations are a2 y2 y — b2x2x′ = a2 62 the tangent.

a2 yy - b2 x x′ = 0

188. The sq. on C D = the rectangle SP, HP 189. If P F be drawn perpendicular on CD, then

the conjugate

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190-2. If a and be the tangents of the angles which a pair of supplemental

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plemental chords. Conjugate diameters are parallel to sup. chords 193. There are no equal conjugate diameters in general. In the equilateral hyperbola they are always equal to each other 194-6. The Asymptotes. The equation to the asymptote is the equation to the curve, with the exception of the terms involving inverse powers of x. Curvilinear asymptotes

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197. The hyperbola is the only one of the lines of the second order that has a rectilineal asymptote

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198. Method of reducing an equation into a series containing inverse powers of a variable. The asymptotes parallel to the axes

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199. Discussion of the equation b x y +ƒ=0

200. Referring the curve to its centre and axes, the equations are

a2 y2 — b2 x2 — — a2 b2, the curve,

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201. In the equilateral or rectangular hyperbola (y2 - x2 = — ao) the angle between

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202, 3. Asymptotes referred to the vertex of the curve; a line parallel to the asymptote cuts the curve in one point only

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Art.

204. Examples of tracing hyperbolas, and drawing the asymptotes

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206. Reduction of the general equation of the second order to the form xy=k2 a (tan. 6)2 + b tan. 6 + c = 0

207. To find the value of b

209. Examples. If ca, the curve is rectangular

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211. Given the equation xy=k, to find the equation referred to rectangular axes, and to obtain the lengths of the axes

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212. From the equation a2 y2 - b2x2 — — a2 b referred to the centre and axes to obtain the equation referred to the asymptotes,

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213. The parallelogram on the co-ordinates is equal to half the rectangle on the

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214. The parts of the tangent between the point of contact and the asymptotes are equal to each other and to the semi-conjugate diameter 215. Given the conjugate diameters to find the asymptotes. If the asymptotes are given, the conjugate to a diameter is given

216. The equation to the tangent referred to the asymptotes

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217, 8. The two parts of any secant comprised between the curve and asymptote are equal. The rectangle S Q, QS' sq. on CD

219. The general polar equation is

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223, 4. The conjugate hyperbola. The locus of the extremity of the conjugate diameter is the conjugate hyperbola. The equation is

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225, 6. The equation to the parabola referred to its axis and vertex is y2= px
227. Difference between a parabolic and hyperbolic branch
228. The equation to the parabola deduced from that to the ellipse referred to its
vertex, by putting AS = m

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229. The principal parameter, or Latus Rectum, is a third proportional to any abscissa and its ordinate. In the following articles 4m is assumed to be the value of the Latus Rectum

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Art.

230. To find the position of the focus

231. The distance of any point on the curve from the focus, S P = x+m 232. The equation to the tangent is

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235. The equation to the tangent at the extremity of the Latus Rectum is

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236. The Directrix. The distances of any point from the focus and directrix are equal 237. The length (Sy) of the perpendicular from the focus on the tangent =√m'r SP: Sy::Sy: SA

238, 9. The locus of y is the axis A Y. The perpendicular Sy cuts the directrix on the point where the perpendicular from P on the directrix meets that line 240. The tangent makes equal angles with the focal distance and with a parallel

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242. The subnormal is equal to half the Latus Rectum :

SGSP, and PG4mr

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243. The parabola has an infinite number of diameters, all parallel to the axis 123 244, 5. Transformation of the equation to another of the same form referred to a

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246. The new equation is y2=px; the new parameter p = 4 SP 247. Transformation of the equation when the position of the new origin and axes

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248. The ordinate through the focus 4SP the parameter at the origin 249. The equation to the tangent

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250. Tangents drawn from the extremities of a parameter meet at right angles in

251. The general polar equation is

(yu sin. 4)2=p (x2+u cos. 0)

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