Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Math 121B, HW4
Problems:
(1) In the gure below, E and F are the foci of the ellipse and P R is the line tangent to
the ellipse at the point P . In this problem you will prove that the tangent line creates
equal angles with the focal radii, i.e. you will p
Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Circle Problem Three
We have seen that a circle with center (a, b) and radius r has the equation
(x a)2 + (y b)2 = r2 .
If we expand the brackets in equation (1) and collect the corresponding terms, we can rewrite
the equation in the form
x2 + y 2 2ax 2by
Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Ellipse Problem Six
Let F be a xed point, let l be a xed line, and let 0 < e < 1. Prove that the set of all
dist(P, F )
= e is an ellipse.
points P in the plane with the property
dist(P, l)
1
Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Ellipse Problem Five
We know that the tangent line to a circle is perpendicular to the radius at the point of
tangency. Is there a similar characterization of the tangent line to an ellipse?
1
Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Circle Problem Two
We have seen that a circle with center (a, b) and radius r has the equation
(x a)2 + (y b)2 = r2 . Samuels math book says that the equation
x2 + y 2 + 2x 4y + 2 = 0
denes a circle. Can Samuel retrieve the center and radius of the circle
Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Math 121B, HW2
(1) Let l be the line dened by the equation y = 2x + 5. Find an equation of a circle C
centered at the origin so that l is tangent to C. Please show your work.
(2) Suppose that C1 and C2 are circles which intersect at the points P and Q. Pr
Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Math 121B, HW3
x2
y2
+ 2 = 1, where a2 > b2 . Give
a2
b
an algebraic proof (i.e. use the equation of the ellipse that we derived in lecture on
Thursday, April 17) that:
(a) E is symmetric about its major axis.
(b) E is symmetric about its minor axis.
(1)
Foundations of Teaching and Learning Mathematics II
MATH 121

Spring 2014
Math 121B, HW1
(1) Prove that a line and a circle intersect in two points or one point or not at all.
(2) Prove that a point is on the perpendicular bisector of a line segment if and only if it
is equidistant from the endpoints of the segment.
(3) Without