a hyperbola if \(A\) and \(C\) have opposite signs.
A system of two quadratic equations may have up to four solutions.
Exercises9.6.3Chapter 9 Review Problems
Exercise Group.
For Problems 1 and 2, decide whether the lines are parallel, perpendicular, or neither.
1.
\(y=\dfrac{1}{2}x+3 \text{;}\)\(\quad x-2y=8\)
Answer.
parallel
2.
\(4x-y=6 \text{;}\)\(\quad x+4y=-2\)
Answer.
perpendicular
3.
Write an equation for the line that is parallel to the graph of \(2x + 3y = 6\) and passes through the point \((1, 4)\)
Answer.
\(y=\dfrac{-2}{3}x+\dfrac{14}{3} \)
4.
Write an equation for the line that is perpendicular to the graph of \(2x + 3y = 6\) and passes through the point \((1, 4)\)
Answer.
\(y=\dfrac{3}{2}x+\dfrac{5}{2} \)
5.
Two vertices of the rectangle \(ABCD\) are \(A(3, 2)\) and \(B(7, −4)\text{.}\) Find an equation for the line that includes side \(\overline{BC}\text{.}\)
Answer.
\(y=\dfrac{2}{3}x-\dfrac{26}{3} \)
6.
One leg of the right triangle \(PQR\) has vertices \(P(-8, -1)\) and \(Q(-2, -5)\text{.}\) Find an equation for the line that includes the leg \(\overline{QR} \text{.}\)
Answer.
\(y=\dfrac{3}{2}x-2 \)
7.
Find the perimeter of the triangle with vertices \(A(-1, 2)\text{,}\)\(B(5, 4)\text{,}\)\(C(1, -4)\text{.}\) Is \(\Delta ABC\) a right triangle?
Answer.
21.59; yes
8.
Find the midpoint of each side of \(\Delta ABC\) from the previous problem. Join the midpoints to form a new triangle, and find its perimeter.
Answer.
10.8
Exercise Group.
For Problems 9–18, graph the conic section.
9.
\(x^2+y^2=9\)
Answer.
10.
\(\dfrac{x^2}{9}+y^2=1\)
Answer.
11.
\(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\)
Answer.
12.
\(\dfrac{y^2}{6}-\dfrac{x^2}{8}=1\)
Answer.
13.
\(4(y-2)=(x+3)^2\)
Answer.
14.
\((x-2)^2+(y+3)^2=16\)
Answer.
15.
\(\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1\)
Answer.
16.
\(\dfrac{(x+4)^2}{12}+\dfrac{(y-2)^2}{6}=1\)
Answer.
17.
\(\dfrac{(x-2)^2}{4}+\dfrac{(y+3)^2}{9}=1\)
Answer.
18.
\((x-2)^2+4y=4\)
Answer.
Exercise Group.
For Problems 19–30
Write the equation of each conic section in standard form.
Identify the conic and describe its main features.
19.
\(x^2+y^2-4x+2y-4=0\)
Answer.
\(\displaystyle (x-2)^2+(y+1)^2=9\)
Circle: center \((2,-1)\text{,}\) radius 3
20.
\(x^2+y^2-6y-4=0\)
Answer.
\(\displaystyle x^2+(y-3)^2=13\)
Circle: center \((0,3)\text{,}\) radius \(\sqrt{13}\)
For Problems 43–48, write and solve an equation or a system of equations
43.
Moia drives 180 miles in the same time that Fran drives 200 miles. Find the speed of each if Fran drives 5 miles per hour faster than Moia.
Answer.
Moia: 45 mph, Fran: 50 mph
44.
The perimeter of a rectangle is 26 inches and the area is 12 square inches. Find the dimensions of the rectangle.
Answer.
12 in by 1 in
45.
The perimeter of a rectangle is 34 centimeters and the area is 70 square centimeters. Find the dimensions of the rectangle.
Answer.
7 cm by 10 cm
46.
A rectangle has a perimeter of 18 feet. If the length is decreased by 5 feet and the width is increased by 12 feet, the area is doubled. Find the dimensions of the original rectangle.
Answer.
7 ft by 2 ft
47.
Norm takes a commuter train 10 miles to his job in the city. The evening train returns him home at a rate 10 miles per hour faster than the morning traintakes him to work. If Norm spends a total of 50 minutes per day commuting, what is the rate of each train?
Answer.
Morning train: 20 mph, evening train: 30 mph
48.
Hattie’s annual income from an investment is $32. If she had invested $200 more and the rate had been 1/2% less, her annual income would have been $35. What are the amount and rate of Hattie’s investment?