23. Or if tangents be drawn from the extremity of each to meet the other produced, the two triangles so formed will be equal in area. 24. If CA, CB be the semi-axes of an ellipse and the rectangle BCAD is completed: shew that if an ellipse similar and similarly situated to the given ellipse be described about it, and any chord DPQR be drawn cutting the first ellipse in P, R, and the second in Q, PQ = QR. 25. Inscribe in a triangle an ellipse with axes parallel to given lines, and in a given ratio. 26. If the elliptic area between the two radii CP, CQ is invariable, prove that the area between the chord PQ and the curve will be so also. 27. PD joins the extremities of conjugate diameters, and CIJ makes a constant angle with PD, meeting it in I, and the parallel tangent in J. Shew that I and J trace similar curves. 28. If P be any point of an ellipse, and AP, A'P produced meet the tangents at A', A in R and S, the tangent at P will bisect AS and A'R. 29. If from an external point O, two straight lines OAP, OQA' be drawn through the vertices of an ellipse APA'Q: if QA, A'P intersect in R, OR is at right angles to the axis major. 30. If TP, TP' be tangents to an ellipse, and PCp be the diameter through P, then P'p is parallel to CT. 31. If TP, TP be tangents to an ellipse from an external point T, TR the diagonal of the parallelogram on TP, TP' and R be on the ellipse, then T will lie on an ellipse similar and similarly situated to the former. 32. If from any point T exterior to an ellipse, a line be drawn parallel to either axis to meet the curve the first or second time in Q, the line bisecting TQ at right angles and that bisecting the tangents from T will meet on the tangent at Q. 33. Find the centre of a given ellipse. 34. Find the axes of a given ellipse. 35. In a given ellipse find the diameter conjugate to a given diameter. 36. If two points of a rod be constrained to move in two fixed lines which intersect at right angles, every other point of the rod will describe an ellipse. 37. If from any point P of an ellipse PQ be drawn to the axis major equal to BC, then if PQ produced either way meets the axis minor in R, PR= AC. 38. If PCP', DCD' are conjugate diameters, and PQ is drawn parallel to the axis major to meet the curve in Q; prove that DQ is parallel to two of the lines joining extremities of the axes of the curve. 39. If NP produced meets the auxiliary circle in Q, prove that GP, CQ produced meet on a circle whose diameter = sum of the axes of the curve. 40. Two ellipses have their axes equal each to each and in the same plane, also their centres coincident, draw the common tangents. 41. If CA, CB be any conjugate diameters of an ellipse and CB be produced to any point B', and an ellipse is described on CA, CB' as conjugate diameters; if the ordinate P'PN be drawn parallel to BC, shew that the tangents to the ellipses at P, will intersect at a point lying on CA produced, and that PN: P'N :: BC : B'C. P' 42. If two ellipses with major axes parallel or at right angles intersect in four points, the opposite sides of the quadrilateral formed by joining the four points will be equally inclined to either axis of either curve. 43. If any system of diameters of an ellipse of a given even number divide it into equal sectors, the sum of the squares on the diameters is the same whatever their directions. 44. The same when the number is odd or even. 45. Prove that PG. Pg = CD3. 46. If CR be the perpendicular from the centre on the tangent at P, and BR, AD be joined; prove that the triangles ACD, RCB are similar. CHAPTER IV. Focal Properties of the Ellipse. 1. We have now to shew that there are two points within the ellipse which, regarding the curve in its relation to the circle, are as it were a divided centre, and also two lines exterior to it which bear a remarkable relation to these points and the curve. 2. Let us premise the following propositions : If the circle inscribed in the triangle VAA' touches the sides at K, S, L; and the circle that touches AA' and VA, VA' produced touches them at H, K', L', then KK' = AA', and ASA'H. also For the perimeter of triangle VAA' =VA+AH+HA'+A'V = VA+AK' + L'A' + VA' = VK' + VL' = 2VK' =VK+KA+AA' + A'L+LV =2VK+2KK': = VK+AS+AA' + A'S + LV=2VK+2AA', Also KK' KA+AK' = AS+ AH, = :. AA'=AH+AS: but AA' = AH+A'H, Also if we produce AA' both ways to meet LK, K'L' produced in X, X', AX will= A'X'. For AK, A'L' being equal to AS, A'H are equal, and being equally inclined to the axis of the cone will have equal projections on it. And the projections of AX, A'X' will equal those of AK, A'L', and are therefore equal: therefore AX, A'X', being parts of the same line with equal projections on the axis of the cone, are equal. 3. Now to return to the construction in Chap. III. § 1. Let AA' be the intersection of the plane of the paper with a plane at right angles to it that cuts the cone VAA' in the ellipse APA'. Inscribe in the triangle VAA' the circle SKL with centre on the axis of the cone, and escribe the circle HK'L' with centre O' also on the axis. Then if we make the circles to revolve about their diameters which coincide with the axis of the cone, they will generate spheres which will touch the cone in the circles KRL, K'R'L'. Every point of each of these circles is equidistant from V, and therefore the distance RR' from one circle to another along a generating line of the cone will be invariable. Also OS, O'H will be at right angles to the cutting plane, which will therefore touch the spheres in S and H, |