or (4) To find the locus of the poles of normal chords of an ellipse. The equation of the normal at any point is The equations (i) and (ii) will represent the same straight line, if therefore, by squaring and adding the two last equations, we have (5) If a quadrilateral circumscribe an ellipse, the line through the middle points of its diagonals will pass through the centre of the ellipse. Let the eccentric angles of the four points of contact of the tangents be a, ẞ, y, d. The equations of the tangents at the points a, ẞ are The co-ordinates of the middle point of the line joining these points of intersection are given by x= acos(a+B) cos 1 (y − 8) + cos(y+d) cos (a – ẞ) b sin § (a +ß) cos § (y −8) + sin 3 (y +8) cos } (a – B) ̧ y 2 cos (y-d) cos(a – B) Therefore the line joining the centre of the ellipse to this point makes with the major axis an angle the tangent of which is b sin (a+ẞ) cos (y − d) + sin † (y+ d) cos † (a − ẞ) The symmetry of the above result shews that the line joining the centre of the ellipse to one of the middle points of the diagonals of the quadrilateral will pass through the other two middle points. EXAMPLES ON CHAPTER VI. 1. If SP, S'P be the focal distances of a point P on an ellipse whose centre is C, and CD be the semi-diameter conjugate to CP; shew that SP. S'P = CD2. 2. The tangent at a point P of an ellipse meets the tangent at A, one extremity of the axis ACA', in the point Y; shew that CY is parallel to A'P, C being the centre of the curve. 3. A point moves so that the sum of the squares of its distances from two intersecting straight lines is constant. Prove that its locus is an ellipse, and find the eccentricity in terms of the angle between the lines. - 4. P, Q are fixed points on an ellipse and R any other point on the curve; V, V' are the middle points of PR, QR, and VG, V'G' are perpendicular to PR, QR respectively and meet the axis in G, G'. Shew that GG' is constant. 5. A series of ellipses are described with a given focus and corresponding directrix; shew that the locus of the extremities of their minor axes is a parabola. 6. PNP' is a double ordinate of an ellipse, and Q is any point on the curve; shew that, if QP, QP' meet the major axis in M, M' respectively, CM. CM'CA. 7. Lines are drawn through the foci of an ellipse perpendicular respectively to a pair of conjugate diameters and intersect in Q; shew that the locus of Q is a concentric ellipse. 8. The tangent at any point P of an ellipse cuts the equi-conjugate diameters in T, T"; shew that the triangles TCP, T'CP are in the ratio of CT2 : CT"2. 9. If CQ be conjugate to the normal at P, then will CP be conjugate to the normal at Q. 10. If P, D be extremities of conjugate diameters of an ellipse, and PP', DD' be chords parallel to an axis of the ellipse; shew that PD' and P'D are parallel to the equiconjugates. 11. If P, D are extremities of conjugate diameters, and the tangent at P cut the major axis in T, and the tangent at D cut the minor axis in T"; shew that TT" will be parallel to one of the equi-conjugates. 12. QQ' is any chord of an ellipse parallel to one of the equi-conjugates, and the tangents at Q, Q' meet in T; shew that the circle QTQ' passes through the centre. 13. In the ellipse prove that the normal at any point is a fourth proportional to the perpendiculars on the tangent from the centre and from the two foci. 14. Two conjugate diameters of an ellipse are drawn, and their four extremities are joined to any point on a given circle whose centre is at the centre of the ellipse; shew that the sum of the squares of the lengths of these four lines is constant. 15. PNP' is a double ordinate of an ellipse whose centre is C, and the normal at P meets CP' in O; shew that the locus of O is an ellipse. 16. If the normal at any point P cut the major axis in G, shew that, for different positions of P, the locus of the middle point of PG will be an ellipse. 17. A, A' are the vertices of an ellipse, and P any point on the curve; shew that, if PN be perpendicular to AP and PM perpendicular to A'P, M, N being on the axis AA', then will MN be equal to the latus rectum of the ellipse. 18. Find the equation of the locus of a point from which two tangents can be drawn to an ellipse making angles 01, მ., with the axis-major such that (1) tan 0, +tan 0, is constant, (2) cot 0, + cot 0, is constant, and (3) tan 6, tan 6, is constant. 1 2 1 2 2 19. The line joining two extremities of any two diameters of an ellipse is either parallel or conjugate to the line joining two extremities of their conjugate diameters. 20. If P and D are extremities of conjugate diameters of an ellipse, shew that the tangents at P and D meet on the ellipse + = 2, and that the locus of the middle point of a2 b2 21. A line is drawn parallel to the axis-minor of an ellipse midway between a focus and the corresponding directrix; prove that the product of the perpendiculars on it from the extremities of any chord passing through that focus is constant. 22. If the chord joining two points whose eccentric angles are a, ẞ cut the major axis of an ellipse at a distance d from the B d-a centre, shew that tan tan where 2a is the length d+a' of the major axis. α 2 2 = 23. If any two chords be drawn through two points on the axis-major of an ellipse equidistant from the centre, shew that В a tan tan tantan 2 2 8 2 angles of the extremities of the chords. 24. If S, H be the foci of an ellipse and any point A be taken on the curve and the chords ASB, BHC, CSD, DHE..... be drawn and the eccentric angles of A, B, C, D,... be 01, 0, 0, 0, 0. Ꮎ . 0. 0 tan tan = 2 Ꮎ.. 2 25. Shew that the area of the triangle formed by the tangents at the points whose eccentric angles are a, ẞ, y respectively is ab tan (ẞ − y) tan 1 (y − a) tan 1 (a — ß). 26. Prove that, if tangents be drawn to an ellipse at points whose eccentric angles are 1, ò̟,, ., the radius of the circle circumscribing the triangle so formed is P, q, r being the length of the diameters of the ellipse parallel to the sides of the triangle, and a, b the semi-axes of the ellipse. 27. From any point P on an ellipse straight lines are drawn through the foci S, H cutting the corresponding directrices in Q, R respectively; shew that the locus of the point of intersection of QH and RS is an ellipse. 28. If P, p be corresponding points on an ellipse and its auxiliary circle, centre C, and if CP be produced to meet the auxiliary circle in q; prove that the tangent at the point on the ellipse corresponding to q is perpendicular to Cp, and that it cuts off from Cp a length equal to CP. 29. If P, Q be the points of contact of perpendicular tangents to an ellipse, and p, q be the corresponding points on the auxiliary circle; shew that Cp, Cq are conjugate diameters of the ellipse. 30. From the centre C of two concentric circles two radii CQ, Cq are drawn equally inclined to a fixed straight line, the first to the outer circle, the second to the inner: prove that the locus of the middle point P of Qq is an ellipse, that PQ is the normal at P to this ellipse, and that Qq is equal to the diameter conjugate to CP. 31. If w is the difference of the eccentric angles of two points on the ellipse the tangents at which are at right angles, prove that ab sin w=λu, where A, μ are the semi-diameters parallel to the tangents at the points, and a, b are the semi-axes of the ellipse. 32. Two equal circles touch one another, find the locus of a point which moves so that the sum of the tangents from it to the two circles is constant. 33. Prove that the sum of the products of the perpendiculars from the two extremities of each of two conjugate diameters on any tangent to an ellipse is equal to the square of the perpendicular from the centre on that tangent. |