Conics: The definitions of
the ellipse and hyperbola will the definitions based on sum or difference of
distances to the two foci, as given in Ogilvy, GTC, or Lab 4.
The construction given in each
of these places that constructs a conic from a circle with center F1 and a
point F2 inside (for ellipses) and outside (for hyperbolas) is sometimes called
the
Problem 1 (from Lab 4)
a)
Explain carefully why the
b)
Also explain why the perpendicular bisector in the
construction is a tangent line of the hyperbola.
Finally, in a figure, draw a
circle c with center F1 and a point F2 outside the circle.
c)
Construct the points P1 and P2 where the hyperbola
constructed from this data intersects line F1F2.
d)
Construct the two lines that are asymptotes of the
hyperbola.
Problem 2 (extension of Lab 4)
A weakness of the
Draw two points F1 and F2 and somewhat randomly draw a point P. The task in this problem will be to construct
the ellipse e with foci F1 and F2 that passes through P and also the hyperbola
h with foci F1 and F2 that passes through P.
There is a discussion of this in Ogilvy that may help.
a)
Construct the tangent line to the ellipse e at P.
b)
Construct a circle c1 with center F1 so that the
director circle construction with this circle and F2 will construct the ellipse
e through P.
Make it clear by words or labels
what you have done.
c)
Construct the tangent line to the hyperbola h at P.
d)
Construct a circle c2 with center F1 so that the
director circle construction with this circle and F2 will construct the
hyperbola h through P.
Make it clear by words or labels
what you have done.
e)
Explain why e and h are orthogonal at P.
Problem 3 (from Friday)
In class Friday we saw that given
a triangle ABC, one can construct circles a, b, c with centers A, B, C so that
the 3 circles are tangent externally.
The method was to construct 3 points on the sides of triangle ABC that
are the points of tangency of the incircle, in other words the feet of the 3
perpendiculars to the sides from the incenter, the
point of concurrency of the interior angle bisectors.
For this problem, also draw a
random triangle ABC and extend the sides to lines.
a)
Construct 3 circles a1, b1, c1
tangent to each other and with centers A, B, C but with one of the circles
containing the other two.
Hint: Try constructing on of
the excircles
of ABC (one of the other 3 circles besides the incircle
that are tangent to lines AB, BC, CA.
b)
Explain what you did.
c)
Once you have done the construction, explain why it
works.
Problem 4 (more tangent circles)
Draw a circle c with center O
and diameter AB. Construct circles c1
and c2 with diameters OA and