Math 1321- Test II Review – Spring 2004
This Test Review – should now be more complete and similar to the exam on Friday – than the one I gave you previously. Be aware of all problems – but you will only see a few of them.
1. A map depicts a house and a gas distribution center. If we construct a rectangular coordinate system with the origin at the
gas distribution center, the house is located at ( - 4, 5). The company is trying to relocated the headquarters to a point
that we label as ( 2, 0 ). If we construct a new coordinate system at the location of the new headquarters, then what are
the new coordinates of the home (house) ?
2. To what point must the origin be translated to so that
the equation (x +2)2 = 4y – 8 will be free of all constant terms and one first degree term ?
3. What does the equation (x – 2)2 + ( y +3 )2 = 9 look like ( new variables ) if the origin is moved to the point (2, -3) ?
4. The equation x2 + y2 + 2x – 4y – 8 + k (x2 + y2 – 2x – 4y + 10 ) = 0 represents a family of circles that pass through the
intersection of two circles . What are the two circles ?
5. Find the value of k of the member of the family of lines that pass through the intersection of the lines
2x – y + 3 = 0 and 2x – 3y + 4 = 0 and has slope 2.
6. Find the intersection of
a) 2x – y = 3
3x + 4y = 5
b) x2/4 - y2/1 = 1 and x2/1 – y2/9 = 4
c) x2 = 4y and y2 = -8x
7. Find the (a) family of lines that
a) pass through the point ( 2, -1)
b) have slope 2
8. Describe the following family of lines.
a) y = mx + 5
b) y – 2 = m (x + 5 )c)
9. Find the directed distance between the lines
a) 12x – 5y = 2 and 12x – 5y = 10
10. Find the area of
a) the parallelogram with base of length 12 and bases on the lines 3x – 4y = 3 and 3x – 4y = 12
Area = base • ht
b) the triangle with base of length 20 and a vertex at ( 2, 3), base on the line 4x + 3y = -3
11. Find the equation of the acute angle bisector of the angles created by the intersection of
4x – 3y = 4 and 12x – 5y = 12
12. Find the equation of the circle that
a) is tangent to the line 12x + 5y = 3 and has a center at the point ( 3, 1)
b) has endpoints of diameter at ( 3, -2) and ( 7, 1 ) (find the center and the radius only)
c) center on the line 4x – y = 3 and is tangent to the line 4x + 3y = 1 at the point (1, -1) ( find the center only)
d) is tangent to both set of axes in quadrant two and passes through the point ______. Keep in mind that a point in
quadrant two is of the following form ( -x, y), x, y > 0 but the radius is always positive. There are two possible
answers. Just find the radius in each case.
e) center at (3, -4) and passes through (-1, 0).
13. Find the equation of the parabola that
a) passes through the point (3, 2) , vertex at (2, -1 ) , axis of symmetry parallel to x axis.
( I just want the length of l.r.)→______
b) directrix y = - 4, and Focus at F ( 2, 6 )
equation: → ______
c) directrix of length 24, opens to the left with focus at F(2, 0 )
equation: →______
d) passes through the points ( 0, 0), (2, - 3), and ( 0, - 5)
15. Find the equation of the ellipse that
a) has l.r. of length (2b2/a ) 2, with vertices at ( -2, -3 ) and ( 6, -3 )
b) endpoints of minor axis at ( 4, 1) and ( 8, 1 ) with focus at ( 6, 6)
c) eccentricity e = c/a = 4/5 with ends of minor axis at (-2,3) and (-2, -3) ______
d) With vertex at the origin, sum of the distances is 16, passing through the point ( 0, 6), horizontal major axis
24. Find the equation of the hyperbola
a) with major axis of length 12 and parallel to y-axis, minor axis of length 20, vertex at ( 2, -3 ) ,
b) Vertex at (2, 4) and ( -8, 4) focus at ( -10, 4 )
25. Prove y = ax2 , where a <0 is increasing on the interval ( - ∞, 0)
26. If a curve C is symmetric with respect to the x-axis and the origin, then it must be symmetric to the y-axis.
Prove:
27. Pay attention to quizzes 8 , 11, and 12, and page 65 in notes – as well as other application type problems.
Do not forget HW problems -
27. (Whispering Gallery): Any sound from one focus of an ellipse reflects off the ellipse directly back to the other focus.
Suppose a whispering gallery has the shape of a semi-ellipse. The distance between the foci is 24 m while the height of
the gallery at its center is 5m. Find the distance that sound travels as it leaves one focus and travels to the other.
Solution: Think of it as an ellipse with c = 12 ( 24/2) , b = ht of gallery = 5.
Looking for 2a = from FtoP and from PtoF/ = sum of the distances
b2 = a2 – c2 → 25 = a2 – 122 → a2 = 25 + 144 = 169 → a = 13 →2a = 26meters
28 A field is in the shape of an ellipse with major axis of length 185 meters and minor axis of length 155 meters. A track 1
meter wide surrounds this field. Find the area of the track.
Note: an ellipse with axis of length 2a and 2b has area A = abπ
Same center:
small ellipse: a = 185, b = 155larger ellipse: a = 187/2 ( 1meter on each side)b= 157 /2
Area of track = Area.arge - Areasmall = ( 187/2)(157/2) π - (185/2)(155/2)π = ______
29. An exhibition tent is in the form of a cylindrical surface with each cross-section a semi-ellipse having base 20 feet and
height 8 feet. How close to either side of the tent can a person 5 feet tall stand straight up ?
ans. the equation is given by : a = 10, b = 8 → x2/ 100 + y2/64 = 1 .
if the person is 5 feet tall , then y = 5, find x. So, let x = y = 5,
solve for x. x2/100 = 1 – 25/64 → x2 = 100 ( 64 – 25)/64 → x = 7.8 (about ),
he can stand 10-7.8 = 2.2 feet away from the edge.
30. A TV satellite antenna consists of a parabolic dish wit the receiver placed at its focus. The dish can be described by
rotating the parabola y = about its axis of symmetry, where -6 ≤ x ≤ 6 and x is measured in feet. How deep is
the dish, and where should the receiver be placed in relation to the bottom of the dish.
solution: the parabola is at its deepest when x is as large as it can be ( 6). y = (1/12 ) ( 6)2 = 3 feet.
rewrite the equation → x2 = 12 y , this tells us that 4a = 12 →a = 3
Now we know to put the receiver 3 units above the origin ( the lowest point )
31. I skipped the last problem – too much work to write up the solution.
32. Here are the examples from quiz 11 ( I did not see anything different in quiz #8 than what is already on this review )
1) If f , f(x) , represents a constant function, then f(x) must be nondecreasing and nonincreasing.
Pf.
Begin with function f being a constant function, f(x) = c. Prove that it is nondecreasing and nonincreasing.
Let x1 and x2 be given with x1 < x2.
a) If f is increasing, then f(x1) < f(x2), but f(x1) = ____ and f(x2) = ____. So, f(x1) ______f(x2)
which means f can not be increasing →f is nonincreasing. Also,
b) If f is decreasing, then f(x1) > f(x2), but we know that f(x1) = _____ and f(x2) = ______
f decreasing means that f(x1) > f(x2), but in this case we have ______
We conclude that f is
2) Suppose that a pipe leans against a wall and a cross section has the appearance below. Find the equation of the circle
represented by the cross section. The circle is tangent to both sets of axes and is of radius 4. The line L has slope
crosses the y-axis at y=12 and the x axis at x = 16.
(Similar but easier than #35 on page 75)C: ______
3) Find an angle created by the intersection by the intersection of the lines = ______(Short and Sweet)
Should not need more space than what is provided
4x – 3y = 4
3x + 4y = 1
4) Find the acute angle bisector created by the intersection of the lines
equation1: 3x – 4y = 3
equation2: 4x – 3y = 42
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