WORKSHEET 6
1. Instructions
I will collect any work you want to turn in on February 8th. You are not expected to
turn in all the problems, but you should choose the problems you think will most benefit
you and work on those.
2. More hyperbolas
Problem 1. Find the vertices, foci, and asymptotes of the following hyperbolas and sketch
them:
(1)
y2
x2
−
= 1;
16
36
(2)
9x2 − 4y 2 = 36;
(3)
y 2 − 4x2 − 2y + 16x = 31.
Problem 2 (Harder). Use the definition of a hyperbola to derive the equation
x2
y2
−
=1
a2
b2
for a hyperbola with foci (±c, 0) and vertices (±a, 0).
Problem 3 (Harder). Show that the function defined by the upper branch of the hyperbola
y 2 /a2 − x2 /b2 = 1 is concave upward [hint: implicit differentiation].
3. Calculus on conic sections
Problem 4.
(1) Show that the equation of the tangent line to the parabola y 2 = 4px
at the point (x0 , y0 ) can be written as
y0 y = 2p(x + x0 ).
(2) What is the x-intercept of this tangent line? Use this fact to draw the tangent
line.
Problem 5. Sketch the parabola y 2 = 4px (you can assume p > 0 for this problem). If
you fix any point on the directrix, for example, (−p, a), there are exactly two tangent lines
to the parabola passing through this point. Show analytically that these two tangent lines
are perpendicular.
Date: February 6th, 2012.
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WORKSHEET 6
4. 3-Dimensional coordinates
Problem 6. Suppose you start at the origin, move along the x-axis a distance of 4 units
in the positive direction, and then move downward a distance of 3 units. What are the
coordinates of your position [hint: to understand ’downward’, look down - along which
coordinate axis are you looking?].
Problem 7. Sketch the points (0, 5, 2), (4, 0, −1), (2, 4, 6) and (1, −1, 2) on a single set
of coordinate axes.
Problem 8. Which of the points P = (6, 2, 3), Q = (−5, −1, 4) and R = (0, 3, 8) is closest
to the xz-plane? Which point is closest to the yz-plane?
Problem 9. What does the equation x = 4 represent in R2 ? What does it represent in
R3 ? Illustrate with sketches.
Problem 10. Determine if the following sets of points lie on a straight line:
(1) A = (2, 4, 2), B = (3, 7, −2) and C = (1, 3, 3);
(2) P = (0, −5, 5), Q = (1, −2, 4) and R = (3, 4, 2).
Problem 11 (Harder). Sketch the surface that has equation z = x2 +y 2 [hint: does x2 +y 2
remind you of something?]