ST JOHN'S COLLEGE. DEC. 1836. (No. VII.) 1. In equal circles, angles whether at the centres or circumferences have the same ratio as the arcs which subtend them. 2. Every solid angle is contained by plane angles, which together are less than four right angles. 3. If two pairs of common tangents be drawn to two unequal circles, and 2a, 2a' be the angles which the two of each pair make with each other; then 4. A, B, C, D are four points in order in a straight line, find a point E between B and C such that AE.EB=ED.EC by a geometrical construction. 5. If from the centre of a rectangular hyperbola a line be drawn through the point of intersection of two tangents; and if o and p' be the angles which this line and the chord joining the points of contact, respectively make with the real axis; then will tan & . tan o' = 1. 6. There are any number of ellipses having a common centre, and their axes majores in the same position. Shew analytically that if all the ellipses be twisted through the same angle in the same direction, the loci of the intersections of each ellipse with its original position, are two straight lines whose equations are 7. Two given unequal circles touch each other externally; shew that the locus of the centre of the circle which always touches the other two is a hyperbola. Find the axes and eccentricity, and shew what the figure becomes when the given circles are equal. 8. A vessel whose outward figure is a paraboloid of revolution, is required to be of equal thickness throughout; find the figure of the interior surface. 9. In an ellipse, if through the foci S and H, chords PS'P', and QHQ' be drawn parallel to any pair of conjugate diameters, shew that SP.SP' + HQ . HQ' = b2 + 12 where b and are respectively the semi-axis minor, and semi-latus rectum. 10. In any circle draw a chord AB: from the middle point of the lesser segment draw any line cutting AB in C and meeting the circumference in D; join AD and take AP AC; find the locus of P. 11. Round a given ellipse circumscribe a rhombus ; about this rhombus circumscribe a second ellipse, and so on for n times; prove that all the ellipses are similar, and find the sum of the areas of the n ellipses. 12. An ellipse has a square described touching it at the extremities of the minor axis: an ellipse upon the same axismajor circumscribes the square. This ellipse is dealt with in the same manner as before, and the operation is continued till there are altogether n + 1 ellipses; prove that if the original eccentricity the last ellipse becomes a circle. n √√ n + 1 13. Given the equation Ay2+ Bxy + Cx2 + D = 0 to be the equation to the hyperbola; find the position of the asymptotes, and the equation to the hyperbola referred to them as axes. 14. Find the axes and position of the curve represented by the equation y2 − 2xy + 3x2 + 2y – 4x ~ 3 − 0. SOLUTIONS TO (No. VII.) 1. EUCLID, Prop. 33, Book vi. 2. Euclid, Prop. 11, Book x1. 3. If C, C' (fig. 56) be the centres of the two circles; DD'T, ET'E' common tangents to the circles, meeting CC' in T, T' respectively; join CD, C'D', CE, C'E'; then if L CTD = a, LET'C' a', we have = 4. Take any point F (fig. 57) not in AB; about the triangle ABF describe a circle ABFG, and about the triangle DCF describe a circle DCFG. Let the circles intersect each other in G; join GF and produce it to meet AD in E; then EA. ER EF.EG EC.ED. 5. Let h, k be the co-ordinates of the point of intersection of the two tangents; then tan ; and the equation k h to the line joining the points of contact, when tangents are drawn to the hyperbola from the point h, k, is and if the hyperbola be rectangular, tan o tan o' = 1. 6. Taking the centre for the origin, the polar equation to the ellipse is p2 = b2 1 – e2 cos3 o ; and the polar equation to the same ellipse when the axis-major is twisted through an points of intersection of the ellipses in the two different positions, cos* = cos2 († − 0), both which values are independent of the eccentricity and magnitude of the axes. Hence every corresponding pair of ellipses will intersect each other in two straight lines passing axis-major; or the equations to the two lines will be to the 7. Let S, H (fig. 58) be the centres of the two circles whose radii are r, r'; P the centre of a circle touching both circles: then SP HP = SQ - HR HR = r = r-r', and is constant, whose foci are S, H. If 2a, or the locus of P is a hyperbola 26 be the axes of the hyperbola, When ', SP - HP = 0, or P lies in the straight line. which is drawn perpendicular to SH bisecting it; therefore the locus of P in this case becomes the common tangent. 8. Let be the angle which the normal PG (fig. 59) to the parabola makes with the axis AG; X, Y the co-ordinates of P; in PG take PQ = b, then the locus of Q will be a curve which by its revolution round AQ will form the inner surface of the vessel. Since the subnormal 2a, tan .. Y = 2a tan 0, X Y 2 a = a tan2; and b cos30 - (ir + a) cos2 0 + α = Multiply (2) by b and subtract from (1); .. b (x + a) cos2 8 + (y2 - 4ax - b3) cos 0 + 3ab = 0. (3) Multiply (2) by 46 and subtract (1), or 3b2 cos2 0 – 4b (x + a) cos 0 — (y2 4 ax b2) = 0. 0. (4) Multiply (3) by 3b, and (4) by (x + a) and subtract ; ... {3b (y2 - 4ax - b2) + 4b (x + a)} cos 0 +9ab2 + (x + a) (y2 - 4 ax - b2) = 0. Multiply (3) by y2 -4ax-b3, and (4) by 3ab and add; .. {b (x + a) (y2 − 4 a x − b2) + 9ab3} cos 0 Hence + (y2 − 4 ax − b3)? – 12 a b2 (x + a) = 0. {9 ab2 + (x + a) (y2 — 4ax − b2)}2 = { 3 (y2 — 4 ax − b3) + 4 (x + a)2 } { (y2 − 4a x − b2)2 – 12 ab2 (x + a)}, which is the equation required. 9. If Aa (fig. 60) be the axis-major, and a' be the semidiameter parallel to PSP', we have |