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William Henry Drew.

A geometrical treatise on conic sections with numerous examples for the use of schools and students in the universities : with an appendix on harmonic ratio, poles and polars, and reciprocation online

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Online LibraryWilliam Henry DrewA geometrical treatise on conic sections with numerous examples for the use of schools and students in the universities : with an appendix on harmonic ratio, poles and polars, and reciprocation → online text (page 7 of 11)
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the axis again in D, and a tangent is drawn to the circle at
D : if two tangents be drawn to the circle from any point in
the conic they will intercept between them a constant length
of the former tangent.

117. If the lines which bisect the angles between pairs of
tangents to an ellipse be parallel to a fixed straight line,
prove that the locus of the points of intersection of the
tangents will be a rectangular hyperbola.

118. An hyperbola, of given eccentricity, always passes
through two given points; if one of its asymptotes always
pass through a third given point in the same straight line
with these, prove that the locus of the centre of the hyperbola
will be a circle.

119. A, P and B, Q are points taken respectively in two
parallel straight lines, A and B being fixed, and P, Q variable.
Prove that if the rectangle APBQ be constant, the line PQ
will always touch a fixed ellipse or a fixed hyperbola, ac-
cording as P and Q are on the same or opposite sides of
AB.

120. If two plane sections of a right cone be taken, having
the same directrix, the foci corresponding to that directrix lie
on a straight line which passes through the vertex.

121. Give a geometrical construction by which a cone may
be cut so that the section may be an ellipse of given eccen-
tricity.



CONIC SECTIONS. 139

122. Given a right cone and a point within it, there are
but two sections which have this point for focus ; and the
planes of these sections make equal angles with the straight
line joining the given point and the vertex of the cone.

123. If the curve formed by the intersection of any plane
with a cone be projected upon a plane perpendicular to the
axis, prove that the curve of projection will be a conic section
having its focus at the point m which the axis meets the
plane of projection.

124. If F be the point where the major axis of an elliptic
section meets the axis of the cone, and be the centre of the
section ; prove that

CF : C'S :: AA' : AO + A'O,

being the vertex of the cone.



SECOND SEPJES.

1. If OQ, OQ' be tangents to a parabola, and OF" drawn
parallel to the axis meet the directrix in K and QQ' in l r ,
and QQ' meet the axis in X, OKXS shall be a parallelo-
gram.

2. G is the foot of a normal at a point P of a parabola, Q
is the middle point of SG, and X is the foot of the directrix,
prove that

QX 2 - QP- 2 = 4AS\

3. Through any point tangents are drawn to a parabola ;
and through straight lines are drawn parallel to the normals
at the points of contact, prove that one diagonal of the
parallelogram so formed passes through the focus.

4. Two parabolas have their foci coincident, prove that the
common chord passes through the intersection of the direc-
trices, and that they cut one another at angles which are half
the angles between the axes.



140 CONIC SECTIONS.

5. A circle is drawn through the point of intersection of
two given straight lines and through another given point,
prove that the straight line joining the points, where the
circle again meets the two given straight lines, touches a fixed
parabola.

G. Common tangents are drawn to two parabolas which
have a common directrix and intersect in P, Q ; prove that
the chords joining the points of contact in each parabola are
parallel to PQ ; and the part of each tangent between its
points of contact with the two curves is bisected by PQ
produced.

7. The tangents at two points Q, Q' in the parabola meet
the tangent at P in P, P' respectively, and the diameter
through their point of intersection T meets it in K\ prove
that PR = KB' and that if QM, Q'M/ TN be the ordinates
of Q, Q,' T respectively to the diameter through P, PNis a
mean proportional between PM and PM' .

8. P and p are points on a parabola on the same side of
the axis and PN, pn the ordinates; the normals at P and p
intersect in Q ; prove that the distance of Q from the axis

_ 2 P N. pn{PN + p n)
(latus rectum) 2

Deduce the value of the radius of curvature.

9. Two equal parabolas have a common focus, and from
any point on the common tangent another tangent is drawn
to each ; prove that these tangents are equidistant from the
common focus.

10. From an external point two tangents are drawn to a
parabola, and from the points where they meet the directrix
two other tangents are drawn meeting the tangents from at
A and B.

Prove that AB passes through the focus S, and that OS is
at right angles to AB.



CONIC SECTIONS. 141

11. Given two tangent lines to a parabola and the focus,
show how to determine the eurve.

12. Inscribe in a given parabola a triangle having its sides
parallel to those of a given triangle.

13. Normals at P, p the extremities of a focal chord of a
parabola meet the axis in G and g. The tangents meet in T.
Prove that the circles about the triangles SPG, Spg intersect
on T 8 produced at a point P such that S bisects P T.

14. A circle and parabola touch one another at both ends
of a double ordinate to the parabola, prove that the latus
rectum is a third proportional to the parts into which the
abscissa of the points of contact is divided by the circle either
internally or externally.

15. PQ is a chord of a parabola normal at P; QR is
drawn parallel to the axis to meet the double ordinate PP'
produced in B; then the rectangle contained by PP' and
P' Pi is constant.

16. OP, OQ are two tangents to a parabola, and OM is
drawn perpendicular to the axis ; if R be the point where
PQ cuts the axis, MP is bisected by the vertex.

17. A parabola, whose focus is 8, touches the three sides of
a triangle ABC, bisecting the base EC in D ; prove that
A 8 is a fourth proportional to A D, A B, and A 0.

18. If the chord of contact be normal at one end, the
tangent at the other is bisected by the perpendicular through
the focus to the line joining the focus to the external point.

19. Two parabolas have a common focus, and from any
point on their common tangent are drawn two other tangents
to the parabolas; prove that the angle between them is
equal to the angle between the axes of the parabolas.

20. If tangents be let fall on any tangent to a parabola
from two given points on the axis equidistant from the focus,
the difference of their squares is constant.



142 ccmc SECTIONS.

21. If a quadrilateral be inscribed in a circle, one of the
three diagonals of the quadrilateral passes through the focus
of the parabola which touches its four sides.

22. A chord of a parabola is drawn parallel to a given
straight line, and on this chord as diameter a circle is de-
scribed ; prove that the distance between the middle point of
this chord, and of the chord joining the other two points of
intersection of the circle and parabola will be of constant
lenoth.

23. Through any point P of an ellipse a line is drawn
perpendicular to the radius vector CP meeting the auxiliary
circle in R, R' : prove that RE' is equal to the difference of
the focal distances of the extremity of the diameter conjugate
to CP.

24. An ellipse and parabola whose axes are parallel, have
the same curvature at a point P and cut one another in Q ;
if the tangent at P meet the axis of the parabola in T prove
that PQ is four times PT.

25. A triangle is inscribed in an ellipse so that each side
is parallel to the tangent at the opposite angle : prove that
the sum of the squares on the sides is to the sum of the
squares on the axes as nine to eight.

26. In an ellipse the perimeter of the quadrilateral formed
by the tangent, the perpendiculars from the foci and the trans-
verse axis, will be the greatest possible, when the focal dis-
tances of the point of contact are at right angles to each other.

27. Two given ellipses in the same plane have a common
focus, and one revolves about the common focus, while the
other remains fixed ; prove that the locus of the point of
intersection of their common tangents is a circle.

23. Perpendiculars SY, S r Y' are drawn from the foci upon
a pair of tangents TY, TY' ; prove that the angles STY,
S J TY' are equal or supplementary to the angles at the base
of the triangle formed by joining F, Y' to the centre of the
ellipse.



CONIC SECTIONS. 143

29. With the centre of perpendiculars of a triangle as
centre, are described two ellipses, one about the triangle, and
the other touching its sides : prove that these ellipses are
similar and their homologous axes at right angles.

30. Through a point P of an ellipse a line PDE is drawn
cutting the axes so that the segments PI) and PE are equal
to the two semi-axes respectively ; perpendiculars to the axes
through D and E intersect in ; prove that PO is a normal.

31. If the focal distance SP of an ellipse meet the con-
jugate diameter in E ; then the difference of the squares on
UP and SE will be constant.

32. The chords of curvature through any two points of an
ellipse in the direction of the line joining them are in the
same ratio as the squares of the diameters parallel to the
tangents at the points.

33. From the extremities of the diameter of an ellipse,
perpendicular to one of the equi-conjugate diameters, chords
are drawn parallel to the other. Prove that these chords are
normal to the ellipse.

34. An ellipse of given semi-axes touches three sides of a
given rectangle ; find its centre and foci.

35. If P, Q be any two points on a fixed ellipse, whose foci
are S, //and if JSI\ II Q intersect within the ellipse at R,
prove that two ellipses can be drawn touching each other at
It, the one having 8 for focus and touching the given ellipse
at Q, the other having II for focus and touching the given
ellipse at P.

If the major axis of one of the variable ellipses be given
find the loci of their other foci, and of their point of contact.

36. Find the positions of the foci and directrices of an
ellipse, which touches at two given points P, Q, two given
straight lines PO, QO, and has one focus on the line PQ,
the angle PO Q being less than a right angle.



144 CONIC SECTIONS.

37. If Pp be drawn parallel to the transverse axis of a
given ellipse, meeting the ellipse in P,p, and the circle whose
diameter is the conjugate axis in B, r, then shall

Pp : Br :: QQ f : PP'.

38. Through any point P of an ellipse are drawn straight
lines APQ; A' P R meeting the auxiliary circle in Q, B, and
ordinates Qq, Br are drawn to the transverse axis; prove
that, L being an extremity of the latus rectum

Aq . A'r : At . A'q :: AC 2 : SL 2 .

39. Two ellipses whose axes are equal, each to each, are
placed in the same plane with their centres coincident, and
axes inclined to each other. Draw their common tangents.

40. If S, S' be the two foci of an ellipse ; and $ Y the
perpendicular from S upon any tangent, prove that /S' Y will
bisect the corresponding normal.

41. OB is a diagonal of the parallelogram of which OQ,
OQ', tangents to an ellipse, are adjacent sides : prove that if
R be in the ellipse, will lie on a similar and similarly
situated concentric ellipse.

42. A circle passes through a focus of an ellipse, has its
centre on the major axis of the ellipse, and touches the
ellipse : shew that the straight line from the focus to the
point of contact is equal to the latus rectum.

43. If the focus of an ellipse be joined with the point
where the tangent at the nearer vertex intersects any other
tangent, and perpendiculars be drawn from the other focus on
the joining line, and the last mentioned tangent, prove that
the distance between the feet of these perpendiculars is equal
to the distance from either focus to the remoter vertex.

44. A parallelogram is described about an ellipse ; if two
of its angular points lie on the directrices, the other two will
lie on the auxiliary circle.



CONIC SECTIONS. 145

45. PS is the focal chord of a point on an ellipse ; CR is
a radius of the auxiliary circle parallel to PS, and drawn in
the direction from P to S] SQ is a perpendicular on OR;
shew that the rectangle contained by SP and QR is equal to
the square on half the minor axis.

46. From the extremity P of the diameter PQ of an ellipse
the tangent TPT is drawn meeting two conjugate diameters
in T, T. From P Q the lines PR, QR are drawn parallel to
the same conjugate diameters. Prove that the rectangle under
the semi-axes of the ellipse is a mean proportional between
the triangles P QR and CTT.

47. If CP and CD be equal conjugate diameters of an
ellipse, and the tangent and normal at P meet the major axis
in T and G respectively, prove that TC . TG = 20 P\

48. An ellipse is inscribed in a triangle having its centre
at the centre of the circumscribed circle of the triangle ;
prove that the perpendiculars from the corners of the triangle
on the opposite sides will be normals to the ellipse.

49. Tangents are drawn from any point on the auxiliary
circle to the ellipse ; prove that the line joining one of the
points of contact with one of the foci is parallel to the line
joining the other point of contact with the other focus.

50. From a point P without an ellipse, P Q is drawn
parallel to the major axis, Q being either of the points in
which it meets the curve ; then the straight line bisecting
PQ at right angles, the tangent at Q, and the line which
joins the middle points of PK, PL (the tangents drawn
from P) meet in a point.

51. CP and CD are conjugate semi-diameters of an ellipse ;
PQ is a chord parallel to one of the axes ; shew that DQ is
parallel to one of the straight lines which join the ends of
the axes.

52. A series of ellipses pass through the same point and
have a common focus and their major axes of the same
length ; prove that the locus of their centres is a circle.

What are the limits of the eccentricities of the ellipses ?

L



146 CONIC SECTIONS.

53. A parallelogram is inscribed in an ellipse, and from
any point on the ellipse two straight lines are drawn parallel
to the sides of the parallelogram ; prove that the rectangles
under the segments of these straight lines, made by the sides
of the parallelogram, will be to one another in a constant
ratio.

54 If the tangent and normal in an ellipse meet the axis
in T and G respectively, and Q be the corresponding point of
the auxiliary circle, then

TQ : TP :: BC : PG.

55. POP' is a diameter of an ellipse, CD conjugate to
CP; prove that PD, DP' are inversely proportional to the
diameters which bisect them.

56. If PJFhe one side of a parallelogram described about
an ellipse, having its sides parallel to conjugate diameters,
and the lines joining E, F to the foci intersect in 0, 0', shew
that 0, 0' and the foci will lie on a circle.

57. If the normal at any point of a hyperbola meet the
conjugate axis in g, and S be the focus, Pg will be to Sg in a
constant ratio.

58. In the rectangular hyperbola, if circles be described
passing through the centre of the hyperbola and the points
of intersection of the tangent with the pairs of conjugate
diameters, their centres will lie on a fixed straight line.

59. A chord of a rectangular hyperbola is drawn perpen-
dicular to a fixed diameter and the extremities of the chord
and diameter are joined by four straight lines ; then the line
joining the two fresh points of intersection of these lines will
be parallel to a fixed line.

60. A series of confocal ellipses is cut by a confocal hyper-
bola ; prove that either focal distance of any point of inter-
section is cut by its conjugate diameter with respect to the
particular ellipse in a point which lies on a circle.



CONIC SECTIONS. 147

61. If from any point on the hyperbola a tangent be drawn
to the circle described on the transverse axis as diameter, its
length is equal to the semi-minor axis of the confocal ellipse
through the point.

62. With one focus of a given hyperbola as focus, and any
tangent to the hyperbola as directrix, is described another
hyperbola touching the conjugate axis of the former ; prove
that the two will be similar.

63. Two tangents are drawn to the same branch of a
rectangular hyperbola from an external point ; prove that
the angles which these tangents subtend at the centre are
respectively equal to the angles which they make with the
chord of contact.

64. A circle is described through P, P' , the extremities of
any diameter of a rectangular hyperbola, and cutting the
tangent at P and T: prove that P'T and the tangent to the
circle at P meet on the hyperbola.

Go. A fixed hyperbola is touched by a concentric ellipse.
If the curvatures at the point of contact are equal, the area
of the ellipse will be constant.

66. Two equal circles touch a rectangular hyperbola at a
point 0, and intersect it again in P, Q ; P\ Q' respectively;
prove that the points may be so taken that PP', QQ' each
subtend a right angle at 0, and that the straight lines joining
P', Q' to the centre of the circle OPQ will trisect OP, OQ
respectively.

67. In a central conic given a focus, the length of the
transverse axis, and that the second focus lies on a fixed
straight line ; prove that the conic will touch two fixed
parabolas having the given focus for focus.

68. In a hyperbola a circle is described about a focus as
centre, with radius one-fourth of the latus rectum ; prove
that the focal distances of the points of intersection with the
hyperbola are parallel to the asymptotes.

L 2



148 CONIC SECTIONS.

G9. Prove that the angles subtended at the vertices of a
rectangular hyperbola by any chord parallel to the conjugate
axis are supplementary.

70. A line moves in such a manner that the sum of the
spaces on its distances from two given points is constant ;
prove that it always touches an ellipse or hyperbola, the square
on whose transverse axis is equal to twice the sum of the squares
on the distances of the moving line from the given points.

71. The eccentricity of a hyperbola is 2. A point D is
taken on the axis, so that its distance from the focus 8 is
equal to the distance of S from the further vertex A', Pbein^
any point on the curve, A'P meets the latus rectum in K.
Prove that DK and SP intersect on a certain fixed circle.

72. In a hyperbola if PIT, PK be drawn parallel to the
asymptotes, and a line through the centre meet PH, PK
in R, T, and the parallelogram PRQT be completed, Q is a
point on the hyperbola.

73. If an ellipse and hyperbola have their axes coincident
and proportional, points on them equidistant from one axis
have the sum of the squares on their distances from the other
axis constant.

74. In the hyperbola prove that (see fig. Prop. XXIII.)

MD : PN :: AG : EG :: GN : CM.

75. A parabola and hyperbola have the same focus and
directrix, and SPQ is a line through the focus S to meet the
parabola in P, and the nearer branch of the hyperbola in Q ;
prove that PQ varies as the rectangle contained by 8P and
8Q.

7G. If two hyperbolas have their transverse axes parallel,
and their eccentricities equal, they will have parallel asymp-
totes. Does the converse hold ?

77. PP r is any diameter of a rectangular hyperbola, Q any
point on the curve, PR, P'R' are drawn at right angles to
PQ, P' Q respectively, intersecting the normal at Q in R, R' :
prove that QR and QR' are equal.



CONIC SECTIONS. 149



i



'$. In a hyperbola a line parallel to BC, through the inter-
section of the tangent at P with the asymptote, meets DP,
CA produced in the same point, CD and CP being con-
jugate.

70. From a given point in a hyperbola draw a straight
line such that the segment intercepted between the other
intersection with the hyperbola and a given asymptote shall
be equal to a given line. When does the problem become
impossible 1

80. A tangent is drawn from any point on the transverse
axis of a hyperbola to the auxiliary circle ; if the anglu
between this tangent and the ordinate be called the eccentric
angle, shew that the eccentric angles at the points of inter-
section of an ellipse with two hyperbolas, confocal with itself,
are equal.

81. Two unequal parabolas have a common focus and axes
opposite ; a rectangular hyperbola is described touching both
parabolas and having its centre at the common focus ; prove
that the angle between the lines joining the points of contact
to the common focus is 60°.

82. The tangent and normal at any point of a hyperbola
intersect the asymptotes and axes respectively in four points
which lie on a circle passing through the centre of the hyper-
bola ; and the radius of this circle varies inversely as the
perpendicular from the centre upon the tangent.

83. Prove that the rectangular hyperbola which has for its
foci the foci of an ellipse, will cut the ellipse at the extremities
of the equal conjugate diameters.

84. ABC is a given triangle. CA, CB produced are the
asymptotes of a hyperbola cutting AB in P. Find the
position of P when the sum of the squares on its axes is the
greatest possible.

85. From the point of intersection of an asymptote and a
directrix of a hyperbola a tangent is drawn to the curve ;
prove that the line joining the point of contact with the focus
is parallel to the asymptote.



150 CONIC SECTIONS.

86. Find, the position and magnitude of the axes of a
hyperbola which has a given line for asymptote, passes
through a given point, and touches a given straight line in a
given point.

87. Prove that a hyperbola can be described passing
through the extremities of any two diameters of a given
ellipse, having diameters conjugate to these for its asymp-
totes.

88. P Q is a normal at a point P of a rectangular hyperbola
meeting the curve again in Q ; prove that P Q is equal to the
diameter of curvature at P.

89. P is a point on a hyperbola whose foci are S and //;
another hyperbola is described whose foci are S and P, and
whose transverse axis is equal to SP — 2PII; shew that the
hyperbolas w 7 ill meet only at one point, and that they will
have the same tangent at that point.

90. The tangent at any point on a hyperbola is produced
to meet the asymptotes, thus forming a triangle ; determine
the locus of the point of intersection of the straight lines
drawn from the angles of this triangle to bisect the opposite
sides.

91. The extremities of the latera recta of all conies which
have a common major axis lie on two parabolas.

92. Having given three points, prove that there are four
straight lines such, that with any one of them as directrix,
and. any one of the given points as focus, a conic section may
be described passing through the other two given points.

93. In any conic section if P G, pg, the normals at the ends
of a focal chord intersect in 0, the straight line through
parallel to Pp bisects G<j.

94. The normal at P to a central conic meets the axes in G
and g ; GK and gh are perpendicular to the focal distance
SP; then PK and Pk are constant.

If hi parallel to the transverse axes meet the normal at P

in C, then hi will be constant.



CONIC SECTIONS. 151

95. A focal chord PSQ of a conic is produced to meet the
directrix in K, and KM, KN are drawn through the feet of
the ordinates PM, QN of P and Q.

If KN produced, meet PM produced, in R, prove that
PR is equal to PM.

96. The normals at the extremities of a focal chord PSQ
of a conic intersect in K, and KL is drawn perpendicular to
PQ; KF is a diagonal of the parallelogram of which SK,
SL are adjacent sides ; prove that KF is parallel to the
transverse axis of the conic.

97. With any point on a given circle as focus and a given
diameter as directrix, is described a conic similar to a given
conic ; prove that all such conies will touch the two similar
conies to which the given diameter is a latus rectum.

98. Every conic section passing through the centres of the
four circles which touch the sides of a triangle, is a rect-
angular hyperbola ; and the locus of the centre of this system
of rectangular hyperbolas is the circle circumscribing the
triangle.

99. Find the locus of the centres of plane sections of a
right cone drawn through a fixed point on the axis of the
cone.

100. Shew how to cut a given cone, so that the section may
be a parabola of given latus rectum.

101. If sections of a right curve be made, perpendicular to
a given plane containing the axis, so that the distance
between a focus of a section and that vertex which lies on
the same generating line in the given plane be constant, prove
that the transverse axes, produced if necessary, of all the
sections will touch one of two fixed circles.

102. Find the position of the vertex and axis of a cone of
given vertical angle, in order that a given parabola may be a
section of the cone.



152 CONIC SECTIONS.

103. Two right cones liave a common vertex ; shew how


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Online LibraryWilliam Henry DrewA geometrical treatise on conic sections with numerous examples for the use of schools and students in the universities : with an appendix on harmonic ratio, poles and polars, and reciprocation → online text (page 7 of 11)
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