29(17)
Case 3d.
Suppoſe the given point A to lie in the given circle, whoſe center
is B.
is B.
Then joining AB and continuing it to meet the given circumference in C and
O, biſect AC in E, and from E towards O ſetting off EH = BC the given
Radius, and with A and B Foci and EH tranſverſe Axis deſcribing an Ellipſe
EKH, it will evidently be the Locus required.
O, biſect AC in E, and from E towards O ſetting off EH = BC the given
Radius, and with A and B Foci and EH tranſverſe Axis deſcribing an Ellipſe
EKH, it will evidently be the Locus required.
PROBLEM VI.
Having a right line BC given, and alſo a circle whoſe center is A, to deter-
mine the Locus of the centers of the circles which ſhall be touched both by the
given right line and alſo by the given circle.
mine the Locus of the centers of the circles which ſhall be touched both by the
given right line and alſo by the given circle.
There are three Caſes, but they are all comprchended under one general
ſolution.
ſolution.
Case 1ſt.
Let the given right line be without the given circle, and let it be
required that the circles to be deſcribed be touched outwardly by the given circle.
required that the circles to be deſcribed be touched outwardly by the given circle.
Case 2d.
Let the given right line be without the given circle, and let it be
required that the circles to be deſcribed, be touched inwardly by the given
circle.
required that the circles to be deſcribed, be touched inwardly by the given
circle.
Case 3d.
Let the given right line be within the given circle, and then the
circles to be deſcribed muſt be touched outwardly by the given circle.
circles to be deſcribed muſt be touched outwardly by the given circle.
General Solution.
From the given center A let fall a perpendicular AG to the given line BC,
which meets the given circumference in D [or in Caſes 2d and 3d is produced
to meet it in D] and biſecting DG in F, and ſetting off FM = FA (which is the
ſame thing as making GM = AD the given Radius) and through M drawing
MLK parallel to the given line BC, with A Focus and LK Directrix deſcribe a
Parabola, and it will be the Locus of the centers of the circles required; for
from the property of the Curve FA = FM, and adding equals to equals, or
ſubtracting equals from equals, as the Caſe requires, FD = FG.
which meets the given circumference in D [or in Caſes 2d and 3d is produced
to meet it in D] and biſecting DG in F, and ſetting off FM = FA (which is the
ſame thing as making GM = AD the given Radius) and through M drawing
MLK parallel to the given line BC, with A Focus and LK Directrix deſcribe a
Parabola, and it will be the Locus of the centers of the circles required; for
from the property of the Curve FA = FM, and adding equals to equals, or
ſubtracting equals from equals, as the Caſe requires, FD = FG.
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib