The Solutions of the Geometrical Problems: Consisting Chiefly of Examples in Plane Co-ordinate Geometry, Proposed at St. John's College, Cambridge, from Dec. 1830 to Dec. 1846. With an Appendix, Containing Several General Properties of Curves of the Second Order, and the Determination of the Magnitude and Position of the Axes of the Conic Section Represented by the General Equation of the Second Degree |
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Page 44
... pairs of common tangents be drawn to two un- equal circles , and 2a , 2a ' be the angles which the two of each pair make with each other ; then sin a sin a R ጥ R + r 4. A , B , C , D are four points in order in a straight line , find a ...
... pairs of common tangents be drawn to two un- equal circles , and 2a , 2a ' be the angles which the two of each pair make with each other ; then sin a sin a R ጥ R + r 4. A , B , C , D are four points in order in a straight line , find a ...
Page 87
... tangent to the interior one , and at its intersections with the exterior ellipse , draw tangents to the latter . Find the locus of the intersections of the latter pairs of tangents . 15. Find the locus of the middle points of a system ...
... tangent to the interior one , and at its intersections with the exterior ellipse , draw tangents to the latter . Find the locus of the intersections of the latter pairs of tangents . 15. Find the locus of the middle points of a system ...
Page 93
... tangent at a point a ' , y ' of the first ellipse is x'x y'y 1. ( 1 ) 2 a b ? Let a pair of tangents be drawn from a point X , Y to the second ellipse , then the equation to the line joining the Y y Xx 12 b'a points of contact is + a 1 ...
... tangent at a point a ' , y ' of the first ellipse is x'x y'y 1. ( 1 ) 2 a b ? Let a pair of tangents be drawn from a point X , Y to the second ellipse , then the equation to the line joining the Y y Xx 12 b'a points of contact is + a 1 ...
Page 117
... pair of tangents as axes , is У k X h 2 h k + cxy = 1 , h , k being the portions of the tangents intercepted between the ellipse and their common point of intersection . Find the corresponding equation to the parabola . 4 . Prove ...
... pair of tangents as axes , is У k X h 2 h k + cxy = 1 , h , k being the portions of the tangents intercepted between the ellipse and their common point of intersection . Find the corresponding equation to the parabola . 4 . Prove ...
Page 118
... pair of common tangents to two circles ; and determine within what limits a point must be situated , so that a straight line may be drawn from it cutting both . 12. Two straight lines include a given angle 2a , and from their point of ...
... pair of common tangents to two circles ; and determine within what limits a point must be situated , so that a straight line may be drawn from it cutting both . 12. Two straight lines include a given angle 2a , and from their point of ...
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Common terms and phrases
2c sin² a₁ a²² ABCD angular points asymptotes axes axis axis-major ay² b₁ bisected C₁ chord of contact co-ordinates conic section conjugate conjugate hyperbola constant cos² cosw curve diagonals diameter draw ellipse equal Euclid find the locus fixed point given points given straight lines hence hk g hyperbola inclined inscribed joining the points latus rectum Let A fig line joining m₁ meet middle point n₂ pair of tangents parabola parallel parallelogram pass through three perpendicular plane points of contact points of intersection polar equation polygon position quadrilateral figure radius remaining sides respectively right angles shew Similarly sin w sin² ST JOHN'S COLLEGE t₁ t₂ tangents be drawn tangents drawn three sides touch vertex y₁
Popular passages
Page 54 - If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another ; the remaining sides shall be in a straight line. Let ABC, DCE be two triangles which have the two sides BA, AC proportional to the two CD, DE, viz.
Page 117 - Similar triangles are to one another in the duplicate ratio of their homologous sides.
Page 117 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle,. shall be equal to the square of the line which touches it.
Page 96 - The rectangle contained by the diagonals of a quadrilateral ,figure inscribed in a circle, is equal to both the rectangles contained by i'ts opposite sides.
Page 16 - MAGNITUDES which have the same ratio to the same magnitude are equal to one another ; and those to which the same magnitude has the same ratio are equal to one another.
Page 28 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one part may be equal to the square on the other part*.
Page 28 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.
Page 10 - ... not in the same plane with the first two ; the first two and the other two shall contain equal angles.
Page 87 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.