By Tom Sibley
This survey textual content with a old emphasis helps a number of assorted classes. It comprises crew tasks regarding using know-how or verbal/written responses. The textual content strives to construct either scholars' instinct and reasoning. it really is excellent for junior and senior point classes.
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Additional info for The geometric viewpoint. A survey of geometries
The centre of curvature at any point P on a conic. except when P coincides with the vertex. may be determined as follows : Refer to Fill. 2. Join p. to the focus F . At P draw the normal PNO cutting the axis in N . At N draw NE perp. to NP to intersect in E the line PF produced. At E draw EO perp . to PE to intersect the normal in O. o is the centre of curvature of the conic at the point P. As P approaches the vertex. the points N. 0 and E move towards one another; the student should test this by taking P in several positions approaching V.
Take any Exercise 3, Fi~. -Let C coin- external point P and draw tangents cide with B and draw the line BD, PB and PC. Join BF and CF and which is now a tangent to the conic. show that the angles PFB and PFC Join DF and BF. This is the limiting are equal. Hence: tangents drawn from case of fig. 3, and DF is perp. to any point to a conic subtend equal BF, because BF and FE now coincide. angles at the focus . At B draw the Hence: the angle subtended at the focus normal BG meeting the axis at G. by that part of the tangent intercepted between the conic and the directrix is a Show that ;~ = the eccentricity of right angle.
E. draw tangents at Band C. duce CF to an y point E . Measure the Show that these meet in D 1 on the angles BFD, EFD-they should be directrix, and that D 1F is perp. to BC. equal. Hence : if a straight lin e cut the Hence: tangents at the extremities of directrix in D and the conic in Band a fo cal chord intersect on the directrix. C, and if D, B and C be joined to the Draw other pairs of tangents from focus, then DF bisects the exterior angle points on the directrix (dotted ), and between BF and CF.