9.3E: Parabolas and Non-Linear Systems (Exercises)
Section 9.3 Exercises
In problems 1–4, match each graph with one of the equations A–D.
A. \(y^2 = 4x\)
B. \(x^2 = 4y\)
C. \(x^2 = 8y\)
D. \(y^2 + 4x = 0\)
1. 2. 3. 4.
In problems 5–14, find the vertex, axis of symmetry, directrix, and focus of the parabola.
5. \(y^2 = 16x\)
6. \(x^2 = 12y\)
7. \(y = 2x^2\)
8. \(x = - \dfrac{y^2}{8}\)
9. \(x + 4y^2 = 0\)
10. \(8y + x^2 = 0\)
11. \((x - 2)^2 = 8(y + 1)\)
12. \((y + 3)^2 = 4(x - 2)\)
13. \(y = \dfrac{1}{4}(x + 1)^2 + 4\)
14. \(x = - \dfrac{1}{12}(y + 1)^2 + 1\)
In problems 15–16, write an equation for the graph.
15.
16.
In problems 17-20, find the standard form of the equation for a parabola satisfying the given conditions.
17. Vertex at (2, 3), opening to the right, focal length 3
18. Vertex at (-1, 2), opening down, focal length 1
19. Vertex at (0, 3), focus at (0, 4)
20. Vertex at (1, 3), focus at (0, 3)
21. The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as?\(x^2 = 4y^2\).At what coordinates should you place the light bulb?
22. If we want to construct the mirror from the previous exercise so that the focus is located at (0, 0.25), what should the equation of the parabola be?
23. A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?
24. Consider the satellite dish from the previous exercise. If the dish is 8 feet across at the opening and 2 feet deep, where should we place the receiver?
25. A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 2 feet across, find the depth.
26. If the searchlight from the previous exercise has the light source located 6 inches from the base along the axis of symmetry and the opening is 4 feet wide, find the depth.
In problems 27–34, solve each system of equations for the intersections of the two curves.
27. \(\begin{array}{l} {y = 2x} \\ {y^2 - x^2} = 1 \end{array}\)
28. \(\begin{array}{l} {y = x + 1} \\ {2x^2 + y^2} = 1 \end{array}\)
29. \(\begin{array}{l} {x^2 + y^2} = 11 \\ {x^2 - 4y^2} = 1 \end{array}\)
30. \(\begin{array}{l} {2x^2 + y^2} = 4 \\ {y^2 - x^2} = 1 \end{array}\)
31. \(\begin{array}{l} {y = x^2} \\ {y^2 - 6x^2} = 16 \end{array}\)
32. \(\begin{array}{l} {x = y^2} \\ {\dfrac{x^2}{4} + \dfrac{y^2}{9} = 1} \end{array}\)
33. \(\begin{array}{l} {x^2 - y^2} = 1 \\ {4y^2 - x^2 = 1} \end{array}\)
34. \(\begin{array}{l} {x^2 = 4(y - 2)} \\ {x^2 = 8(y + 1)} \end{array}\)
35. A LORAN system has transmitter stations A, B, C, and D at (-125, 0), (125, 0), (0, 250), and (0, -250), respectively. A ship in quadrant two computes the difference of its distances from A and B as 100 miles and the difference of its distances from C and D as 180 miles. Find the x- and y-coordinates of the ship’s location. Round to two decimal places.
36. A LORAN system has transmitter stations A, B, C, and D at (-100, 0), (100, 0), (-100, -300), and (100, -300), respectively. A ship in quadrant one computes the difference of its distances from A and B as 80 miles and the difference of its distances from C and D as 120 miles. Find the \(x\)- and \(y\)-coordinates of the ship’s location. Round to two decimal places.
- Answer
-
1. C
3. A
5. Vertex: (0, 0). Axis of symmetry: \(y = 0\). Directrix: \(x = -4\). Focus: (4, 0)
7. Vertex: (0, 0). Axis of symmetry: \(x = 0\). Directrix: \(y = -1/8\). Focus: (0, 1/8)
9. Vertex: (0, 0). Axis of symmetry: \(y = 0\). Directrix: \(x = 1/16\). Focus: (-1/16, 0)
11. Vertex: (2, -1). Axis of symmetry: \(x = 2\). Directrix: \(y = -3\). Focus: (2, 1)
13. Vertex: (-1, 4). Axis of symmetry: \(x = -1\). Directrix: \(y = 3\). Focus: (-1, 5)
15. \((y - 1)^2 = -(x - 3)\)
17. \((y - 3)^2 = 12(x - 2)\)
19. \(x^2 = 4(y - 3)\)
21. At the focus, (0,1)
23. 2.25 feet above the vertex.
25. 0.25 ft
27. \((\dfrac{1}{\sqrt{3}}, \dfrac{2}{\sqrt{3}})\), \((\dfrac{-1}{\sqrt{3}}, \dfrac{-2}{\sqrt{3}})\)
29. \((3, \sqrt{2})\), \((3, -\sqrt{2})\), \((-3, \sqrt{2})\), \((-3, -\sqrt{2})\)
31. \((2\sqrt{2}, 8)\), \((-2\sqrt{2}, 8)\)
33. \((\dfrac{5}{3}, \dfrac{2}{3})\), \((-\dfrac{5}{3}, \dfrac{2}{3})\), \((\dfrac{5}{3}, -\dfrac{2}{3})\), \((-\dfrac{5}{3}, -\dfrac{2}{3})\)
35. (-64.50476622, 93.37848007) \(\approx\) (-64.50, 93.38)