. Algebraic geometry; a new treatise on analytical conic sections . ameters of an ellipse, jp, d thecorresponding points on the auxiliary circle to P and D, and pPN, dDMthe ordinates at P and D, prove thatpN =CIV1, and dM = CN. *23, What does the equation si?+i/=cfl represent when the axes areoblique ? *24, If o be the acute angle between the axes of co-ordinates, the semi-axes of the ellipse x^+y^=(? are \/2 c cos = and \^ c sin ^. *25. If e be the eccentricity of the ellipse in the previous question, _/ 2c03tt \^~\1+ cosoy LOCUS PROBLEMS ON THE ELLIPSE. 216. To fimd the locus of the intersec
. Algebraic geometry; a new treatise on analytical conic sections . ameters of an ellipse, jp, d thecorresponding points on the auxiliary circle to P and D, and pPN, dDMthe ordinates at P and D, prove thatpN =CIV1, and dM = CN. *23, What does the equation si?+i/=cfl represent when the axes areoblique ? *24, If o be the acute angle between the axes of co-ordinates, the semi-axes of the ellipse x^+y^=(? are \/2 c cos = and \^ c sin ^. *25. If e be the eccentricity of the ellipse in the previous question, _/ 2c03tt \^~\1+ cosoy LOCUS PROBLEMS ON THE ELLIPSE. 216. To fimd the locus of the intersection of perpendicular tangentsto an ellipse. The straight line y — mx + Ja^ni^ -l- 6^ is a tangent to the ellipse-j-f 1^= 1 for all values of m. If this passes through («!, y{), y^ = ^ -f Ja^rrfi + b^,(y.^-mXiY = a^m^ + ¥,m^(a^ - iCi^) + 2imjyi + b^- y^ = 0. 204 LOCUS PROBLEMS ON THE ELLIPSE. [chap. x. Hence this quadratic for m gives the slopes of the tangentswhich can be drawn from the point (x^, y^ to the this case the tangents are at rt. ^.. Fio. 131. .. if TOiOTj are the roots of this quadratic, ™1™2 = ~ 1J .. x^ + y^ = a^ + b^ is the equation of the circle, whose centre is at the centre of the ellipse, andwhose radius is Ja^ + ¥. This is called the Director Circle. 217. Chords of an ellipse are drawn from one end of the majoraxis ; find the locus of their middle points. Let (flcosO, hsinO) be any point P on the ellipse, A (a, 0) theend of the major axis from which the chords are drawn. Let {x, y) be the co-ordinates of the middle point of PA. 1 / . n\ J 6 sin S a! = §(a + acosS) and y=—^—; • • = cos9 and -r- = 8m0. ART. 218.] LOCUS PEOBLEMS ON THE ELLIPSE. 205 c /2a! — a\^ iv^ Squaring and adding, ( j +-p- = l is the equation of the locus. ^ ——--P^W5 0. & Sin 9) ^ ^-^^^^i^j^.o) 1 c ^ ^ Fio. 132. This may be written (-i) + 2- = l + J2 4 4 Hence we see that the locus is an ellipse whose centre is at the point {-, 0), whose semi
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