Name: ___________________________ Period: _____

Test Review: Conic Sections

PAP

1-2: For the given equations complete the following:

a) Identify the type of conic section, and Write the equation in standard form, b) List all of the important features of the conic section (a, b, center/vertex, radius, focus…etc.), c) Sketch a graph of the conic section on your paper including all of the important features 1) 9𝑥 2 + 16𝑦 2 + 18𝑥 − 64𝑦 = 71

2) 𝑥 2 − 16𝑦 2 − 2𝑥 + 128𝑦 − 271 = 0

3-4: Graph the given information, then write the equation of the conic section in standard form.

3) A circle with center at (−4, 2) and a point on the circle of (3, −1). 4) A hyperbola with a center at (2, 1), vertex at (2, 4), and a focus at (2, 6). 5-8: Identify the conic, put it in standard form, and identify the important characteristics.

5) −3𝑥 2 − 24𝑥 + 𝑦 − 40 = 8

6) 25𝑥 2 + 16𝑦 2 + 150𝑥 − 160𝑦 + 225 = 0

7) 𝑦 2 − 9𝑥 2 + 36𝑥 − 8𝑦 − 20 = 9

8) 𝑥 2 + 𝑦 2 + 10𝑥 − 4𝑦 + 20 = 0

9-12: Write the equation in standard form given the following information.

9) An ellipse with foci at (−5, 1) and (3, 1) and major vertex at (11, 1). 10) A circle with center at (5, −1) and radius of 4. 11) A hyperbola with foci at (3, 1) and (3, −5) and a vertex at (3, 0). 12) A parabola with a focus at (2, 3) and a point on the directrix of (3, 1). 13-14: For the given equations complete the following:

a) Identify the type of conic section, and Write the equation in standard form, b) List all of the important features of the conic section (a, b, center/vertex, radius, focus…etc.), c) Sketch a graph of the conic section on your paper including all of the important features 13) 2𝑥 2 + 2𝑦 2 − 8𝑥 + 4𝑦 − 8 = 0

14) 3𝑥 2 + 12𝑥 + 𝑦 + 12 = 2

15-16: Graph the given information, write the equation of the conic section in standard form.

15) A parabola with vertex at (2, 3) and a focus at (2, 2). 16) An ellipse with foci at (1, 0) and (−1, 0) and minor axis of length of 4.

17-20: Graph the conic section, label all the important features.

17)

(𝑦+6)2 64

−

(𝑥−3)2 35

=1

19) (𝑥 − 5)2 = 12(𝑦 + 4)

18)

(𝑥+2)2 16

+

(𝑦−1)2 36

=1

20) (𝑥 − 2)2 + 𝑦 2 = 100

21-26: Identify the conic, put it in standard form, and identify the important characteristics.

21) 4𝑥 2 + 9𝑦 2 = 36 23) 2𝑥 2 − 4𝑥 − 𝑦 − 2 = −3 25) 2𝑥 2 + 2𝑦 2 + 20𝑥 − 8𝑦 + 38 = −2

22) 𝑥 2 + 9𝑦 2 + 2𝑥 − 18𝑦 − 7 = 1 24) 𝑥 2 − 3𝑦 2 + 6𝑦 − 4 = 1 26) −3𝑦 2 + 𝑥 − 24𝑦 = 48

27-30: Write the equation in standard form given the following information.

27) An ellipse with center at (2, 3), major vertex at (6, 3) and minor vertex at (2, 0). 28) A circle with endpoints of the diameter at (5, 8) and (−3, 2). 29) A hyperbola with a center at (−1, 2), vertex at (−4, 2) and a focus at (−6, 2). 30) A parabola with vertex at (−1, 4), focus at (−1, 6). 31) Some nuclear power plants are in the shape of a hyperboloid, a solid obtained by rotating a hyperbola about its conjugate axis (i.e. not the transverse axis). Suppose such a cooling tower has a base diameter of 200 feet and the radius at its narrowest point, 180 feet above the ground, is 50 feet. If the radius at the top of the tower is 75 feet, how tall is the tower?

32) A suspension bridge is built with its cable hanging between two vertical towers in the form of a parabola. The towers are 400 feet apart and rise 100 feet above the horizontal roadway, while the center point of the cable is 10 feet above the roadway. Find the equations of the parabola and find the height above the roadway of a point 50 feet from the center of the span.

33) A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 4 centimeters from the vertex. If the depth is to be 8 centimeters, what is the diameter of the headlight at its opening?

34) A circular vent pip with a diameter of 6.5 inches is placed on a roof that has a slope of 5/8. The intersection of the vent pipe and the roof is an ellipse. To the nearest hundredth of an inch, what are the lengths of the major and minor axes (the dimensions of the hole)?