| The locus of the point of intersection of two tangents to a parabola
which cut one another at a constant angle is a hyperbola having the
same focus and directrix as the original parabola. | The locus of the point of intersection of two nodal tangent circles
to a cardioid which cut each other at a constant angle is a limaçon
having the same double point and director circle. |
| The sum of the reciprocals of two focal chords of a conic at right
angles to each other is constant. | The sum of any two nodal chords of a limaçon at right angles to each
other is constant. |
| P Q is a chord of a conic which subtends a right angle at the focus.
The locus of the pole of P Q and the locus enveloped by P Q are each
conics whose latera recta are to that of the original conic as √2 : 1
and 1 : √2 respectively. | If P and Q be two points on a limaçon such that they intercept a
right angle at the node, then the locus of the point of intersection
of the two nodal circles tangent at P and Q respectively, is a limaçon
whose latus rectum is to that of the original limaçon as ½√2 : 1. And
the envelope of the circle described on P Q as a diameter is a limaçon,
whose latus rectum is to that of the original limaçon as 1 : ½√2.
|
| If two conics have a common focus, two of their common chords will
pass through the point of intersection of their directrices. | If two limaçons have a common node, two nodal circles passing each
through two points of intersection of the limaçons, will pass through
the point of intersection of their base circles. |
| Two conics have a common focus about which one of them is turned;
two of their common chords will touch conics having the fixed focus for
focus. | Two limaçons have a common node about which one of them is turned;
two of the nodal circles through two of their points of intersection
will envelope limaçons having fixed node for node. |
| Two conics are described having the same focus, and the distance of
this focus from the corresponding directrix of each is the same; if the
conics touch one another, then twice the sine of half the angle between
the transverse axes is equal to the difference of the reciprocals of
the eccentricities. | If two limaçons are described having the same node and base circles
of the same diameter, and if the limaçons touch each other, then twice
the sine of half the angle between the axes of the limaçons is equal to
the difference of the eccentricities. |
| If a circle of a given radius pass through the focus (S) of a given
conic and cut the conic in the points A, B, C, and D; then SA. SB. SC.
SD is constant. | If a circle of a given radius pass through the node (S) of a given
limaçon and cut it in A, B, C, and D; then
1
—————— is constant.
(SA. SB. SC. SD) |
A circle passes through the focus of a conic whose latus
rectum is 2l and meets the conic in four points whose
distance from the focus are
r₁, r₂, r₃, r₄, then
1 1 1 1 2
— + — + — + — = — .
r₁ r₂ r₃ r₄ l | A circle passes through the node of a limaçon whose latus rectum
is 2l, meeting the curve in four points whose distances from the
node are r₁, r₂, r₃, r₄, then
r₁ + r₂ + r₃ + r₄ = 2l. |
| Two points P and Q are taken, one on each of two conics which have a
common focus and their axes in the same direction, such that PS and QS
are at right angles, S being the common focus. Then the tangents at P
and Q meet on a conic the square of whose eccentricity is equal to the
sum of the squares of the eccentricities of the original conics. | Two points P and Q are taken one on each of two limaçons which
have a common node and their axes in the same direction, such that PS
and QS are at right angles, S being the common node. Then the nodal
tangent circles at P and Q intersect on a limaçon the square of whose
eccentricity is equal to the sum of the squares of the eccentricities
of the original limaçons.
|
A series of conics are described with a common latus rectum; the
locus of points upon them at which the perpendicular from the focus on
the tangent is equal to the semi-latus rectum is given by the equation
p = -r cos 2x | If a series of limaçons are described with the same latus rectum,
the locus of points upon them at which the diameter of the nodal
tangent circle is equal to the semi-latus rectum, is given by the
equation
pr = -cos 2x |
| If POP₁ be a chord of a conic through a fixed point O, then will tan ½P₁SO
tan ½PSO be a constant, S being the focus of the conic. | If POP₁ be a nodal circle of a limaçon passing through a fixed point O,
then will tan ½ P₁SO tan ½ PSO be a constant, S being the node. |
| Conics are described with equal latera recta and a common focus.
Also the corresponding directrices envelop a fixed confocal conic.
Then these conics all touch two fixed conics, the reciprocals of whose
latera recta are the sum and difference respectively of those of the
variable conic and their fixed confocal, and which have the same
directrix as the fixed confocal. | Limaçons are described with equal latera recta and a common node.
Also the director circles envelop a fixed limaçon having a common node.
Then these limaçons all touch two fixed limaçons whose latera recta are
the sum and difference respectively of the reciprocals of the variable
limaçon and of the fixed limaçon, and which have the same base circle
as the fixed limaçon. |
| Every focal chord of a conic is cut harmonically by the curve,
the focus, and the directrix. | Every nodal chord of a limaçon is bisected by the base circle. |
| The envelope of circles on the focal radii of a conic as diameters
is the auxiliary circle. | The envelope of the perpendiculars at the extremities of the nodal
radii of a limaçon is a circle having for the diameter the axis of the
limaçon. |
| The straight line which bisects the angle contained by two lines
drawn from the same point in a parabola, the one to the focus, the
other perpendicular to the directrix, is a tangent to the parabola at
that point. | The nodal circle which bisects the angle between the line drawn from
any point on a cardioid to the cusp and the nodal circle through the
point which cuts the director circle orthogonally, is a tangent circle
at that point. |
| The latus rectum of a parabola is equal to four times the distance
from the focus to the vertex. | The latus rectum of a cardioid is equal to its length on the axis. |
| If a tangent to a parabola cut the axis produced, the points of contact
and of intersection are equally distant from the focus. | If a nodal tangent circle cut the axis of a cardioid, the points of
intersection and of tangency are equally distant from the cusp. |
| If a perpendicular be drawn from the focus to any tangent to a
parabola, the point of intersection will be on the vertical tangent. | If a nodal circle be drawn tangent to a cardioid, the diameter
of such circle passing through the cusp will be a common chord of
this circle and another described on the axis of the cardioid as diameter. |
| The directrix of a parabola is the locus of the intersection of
tangents that cut at right angles. | The base circle is the locus of the intersection of nodal circles
tangent to a cardioid, which cut orthogonally. |
| The circle described on any focal chord of a parabola as diameter
will touch the directrix. | The circle described an any nodal chord of a cardioid as diameter
will be tangent to the base circle. |
| The locus of a point from which two normals to a parabola can be
drawn making complementary angles with the axis, is a parabola. | The locus of the point through which two nodal circles, cutting a
cardioid orthogonally, and making complementary angles with the axis,
can be drawn is a cardioid. |
| Two tangents to a parabola which make equal angles with the axis and
directrix respectively, but are not at right angles, meet on the latus rectum. | Two nodal circles tangent to a cardioid which make equal angles with
the axis and latus rectum, respectively but do not cut orthogonally
intersect on the latus rectum.
|
| The circle which circumscribes the triangle formed by three
tangents to a parabola passes through the focus. | If three nodal circles be drawn tangent to a cardioid, the three
points of intersection of these three circles are on a straight line. |
| If the two normals drawn to a parabola from a point P make equal
angles with a straight line, the focus of P is a parabola. | If the two nodal circles cutting a cardioid orthogonally and
pass through the point P, make equal angles with a fixed nodal circle,
the locus of P is a cardioid. |
| Any two parabolas which have a common focus and their axes
in opposite directions intersect at right angles. | Any two cardioids which have a common cusp and their axes
in opposite directions intersect at right angles. |
A number of other theorems on the limaçon and cardioid are given in Professor Newson’s article in this number of the Quarterly, and these need not be repeated here.
BY W. H. CARRUTH.