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 DegreeJ. & J. J. Deighton, 1847 - 263 páginas |
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Página 6
... axes , take AB , BC each = b2 and with centre C and radius J √2a describe a circle AED ; it will pass through A and ... axis of x . ( 2 ) Since y2 - 2xy sec a + x2 0 , we have 2 Y X Y X JAC - 2 sec a y + ( sec a ) 2 = ( tan a ) 3 ; X ...
... axes , take AB , BC each = b2 and with centre C and radius J √2a describe a circle AED ; it will pass through A and ... axis of x . ( 2 ) Since y2 - 2xy sec a + x2 0 , we have 2 Y X Y X JAC - 2 sec a y + ( sec a ) 2 = ( tan a ) 3 ; X ...
Página 7
... axis of a ; hence the angle between them a ( 45 + 2 ) - ( 45 - 2 ) = a . a O respectively 13. Let 0 ( fig . 9 ) be the origin ; AB the straight line meeting the axes of x and y in the points A , B respectively ; draw OP perpendicular to ...
... axis of a ; hence the angle between them a ( 45 + 2 ) - ( 45 - 2 ) = a . a O respectively 13. Let 0 ( fig . 9 ) be the origin ; AB the straight line meeting the axes of x and y in the points A , B respectively ; draw OP perpendicular to ...
Página 11
... axes . 11. The equation to a circle is y2 + x2 = a ( y + x ) ; what is the equation to that diameter which passes through the origin of co - ordinates ? 12. A side of a triangle being assumed as the axis of x , the equations to the ...
... axes . 11. The equation to a circle is y2 + x2 = a ( y + x ) ; what is the equation to that diameter which passes through the origin of co - ordinates ? 12. A side of a triangle being assumed as the axis of x , the equations to the ...
Página 13
... ' be the equations to the two straight lines ; then if 0 , 0 ' be the angles which they respectively make with the axis of x , tan 9 a , tan0 ′ = a ' , = α , and sin ( 0 ′ – 0 ) = cos NO . II . GEOMETRICAL PROBLEMS . 13 DEC . 1831 .
... ' be the equations to the two straight lines ; then if 0 , 0 ' be the angles which they respectively make with the axis of x , tan 9 a , tan0 ′ = a ' , = α , and sin ( 0 ′ – 0 ) = cos NO . II . GEOMETRICAL PROBLEMS . 13 DEC . 1831 .
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... axis of a at a given angle . 11. If from a point without a given circle , two straight lines be drawn touching the circle ; find the equation to the line joining the points of contact . 12. If a , a , be the sides of a right - angled ...
... axis of a at a given angle . 11. If from a point without a given circle , two straight lines be drawn touching the circle ; find the equation to the line joining the points of contact . 12. If a , a , be the sides of a right - angled ...
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Términos y frases comunes
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
Pasajes populares
Página 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.
Página 117 - Similar triangles are to one another in the duplicate ratio of their homologous sides.
Página 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.
Página 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.
Página 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.
Página 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*.
Página 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.
Página 10 - ... not in the same plane with the first two ; the first two and the other two shall contain equal angles.
Página 87 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.