113. Given S' and a chord through S, find S.

114. Two conjugate diameters being given in position and magnitude, determine the axes.

115. The locus of a point which cuts parallel chords of a circle in a given ratio is an ellipse having double contact with the circle.

116. The circle on a focal radius touches the auxiliary circle.

117. The common tangents of the auxiliary circle and the circles on SP, S'P intersect on the ordinate of P.

118. Conjugate diameters meet the directrix at distances from the axis which contain a constant rectangle.

119. A focal chord and its diameter meet the directrix at distances from the axis which contain a constant rectangle.

120. The centre of a focal chord traces a similar ellipse. 121.

The centre of a chord which cuts the axis in a fixed point describes an ellipse.

[53.

122. A straight line equal to the radius of a circle slides with one end on a fixed diameter and the other end P on the convex side of the circumference. Shew that the coordinates of a point Qin the line vary as those of P, and hence that traces an ellipse.

123. Shew that SP - CA varies as the abscissa of P, and that (SP – CA)2 + (CA — SD)2 = CS 2.

[76.

124. The parallelogram described on conjugate diameters as diagonals is of constant area.

125. A diameter of an ellipse varies inversely as the perpendicular focal chord of the auxiliary circle.

126. If the radii CP, CQ be at right angles,

127. The sum of conjugate diameters is a maximum when they are equal.

128. When is the sum of conjugate diameters a minimum ?

129. SQ, S'Q being perpendiculars to conjugate diameters, the locus of Q is a concentric ellipse.

130. The perpendiculars to the axes from the points in which a common diameter meets the two auxiliary circles intersect two and two on the ellipse. [91.

131. The common diameters of equivalent and concentric ellipses are at right angles. 187. 132. If PP', DD' be conjugate diameters, then PD, PD' are proportional to the diameters parallel to them.

133. The sum of conjugate focal chords is constant.

134. A point in a straight line which slides between two fixed straight lines at right angles traces an ellipse.

135. Parallel diameters of similar and similarly situated conics bisect the same systems of parallel chords.

136. With the pole of a chord as centre a circle can be described touching the four focal radii to the ends of the chord.

137. A circle can be drawn through the foci and the intersections of any tangent with the tangents at the vertices.

138. The equi-conjugate diameters coincide in direction with the diagonals of a rectangle formed by the tangents at the ends of the axes.

139. The central perpendicular on the tangent varies inversely as the conjugate diameter.

140. If the tangent and ordinate at Q meet PP' the diameter of abscissæ in T, V, shew by Art. 101 that

141.

TP : TP' = PV : VP.

Shew directly by the method of Art. 103 that

142. If a chord of a conic subtends a constant angle at the focus, its envelope and the locus of its pole are conics having the same focus and directrix.

[108.

143. The vertex of a circumscribed triangle whose base subtends a constant angle at the focus is a conic.

144. Shew by equating the angles of the quadrilaterals SPTP', SPTP' to eight right angles that the external angle between the tangents is equal to half the sum of the angles which their chord of contact subtends at the foci. [115.

145. What is the corresponding theorem when the direction of PP' falls between the foci?

146. Any tangent meets parallel tangents on conjugate di

ameters.

147. If the tangent at P meets parallel tangents in Q, R, the rectangle PQ . PR is equal to CD3.

148. The polar of a point on the directrix passes through S. 149. The radii from a focus to the ends of a diameter make equal angles with the tangents at those points.

150. The straight lines joining the feet and the poles of any two chords subtend equal angles at the focus.

[111.

151. The intercept on a tangent by parallel tangents subtends a right angle at the focus.

152. If s, s' be the reflexions of S, S' with respect to the tangent at P the triangles SPs', sPS' will be equivalent.

153. A parallel to SP from S' meets SY on a circle.

154. Shew that

155. Also that

SY. CD-SP. CB.

SY2 : CB2 = SP: 2CA - SP.

156. The ordinate bisects the angle YNY', and the points Y, N, C, Y' are concyclic.

157. Shew that PG is bisected by SY', and by S'Y; and that it is a harmonic mean to SY, S'Y'.

158. Tangents being drawn from any point on a circle through the foci, shew that the bisectors of the angles between them pass through fixed points.

159. If the tangent and normal meet either axis in T, G, then CG. CT CS2.

160. The bisectors of the angles between the tangents from any point are tangent and normal to the confocals through that

161. In Art. 114 shew that PG. Pg is equal to SP. PS, and deduce the theorems of Art. 131.

[78, 119. 162. If CP be conjugate to the normal at Q, CQ will be conjugate to the normal at P.

163. If the tangent at P and its diameter meet the axis and directrix in T, D, then DT is parallel to PS.

164. Shew how to draw tangents to an ellipse from a point on the auxiliary circle, or from any other external point.

165. The pole of the tangent at P with respect to the auxiliary circle lies on the ordinate of P.

166. If TP, TP' be the tangents in the first case, SP will be parallel to P'T.

167. A circumscribing parallelogram which has two corners on the directrices has the other two on the auxiliary circle.

168. If an ellipse inscribed in a triangle has one focus at the orthocentre, the other focus will be at the centre of the circumscribed circle.

169. If an ellipse slides between two straight lines at right. angles the locus of its centre is a circle.

170. The straight line joining the foci subtends at the pole of a chord half the sum or difference of the angles which it subtends at the extremities of the chord.

171. The portion of a normal chord intercepted between the directrices subtends at the pole of the chord half the sum of the angles which the straight line joining the foci subtends at the

extremities of the chord.

[112.

172. If a chord be produced to meet the directrices, the parts produced will subtend equal angles at the pole.

173. Supplemental chords equally inclined to the curve have their poles on the director circle. What is the corresponding property of the parabola?

174. The sum of two chords thus drawn is constant.

175. The normal at A is equal to L.

[121.

CHAPTER V.

CHORD-PROPERTIES OF THE RECTANGULAR HYPERBOLA.

136. THE hyperbola has been defined as a conic whose eccentricity exceeds unity. When the eccentricity is equal to √2* the hyperbola is called Rectangular, and also Equilateral, for reasons which will appear.

[155.

137. In the next figure, let S be the focus, and X the point in which the axis meets the directrix. Take C in SX produced, such that CX= SX. On the axis measure CA, CA' mean proportionals to CS, CX. It will be seen from Prop. I. that A, A' are the Vertices, or points in which the curve cuts the axis. The point C, which bisects AA', is the centre.

138. DEF. The circle on AA' is called the Auxiliary Circle.

Its diameter BB' perpendicular to AA' is called the Conjugate Axis, and AA' is called the Transverse Axis. These are also, though equal, called the Minor and Major Axis.

140. The square of the semi-latus-rectum is from the definition equal to 2SX, and therefore to CA. Hence the latus-rectum is equal to the axis.

141. Also SA. SA' = CS2 — CA2 – CA2.

* The ratio of the diagonal to the side of a square.

[139.