Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Write the standard-form equation of an ellipse with the given characteristics. Sketch the ellipse.

8. vertices ( − 5 , 1 ) open negative 5 comma 1 close  and (1, 1), focus ( − 3 , 1 ) open negative 3 comma 1 close
9. vertices ( 3 , − 1 ) open 3 comma negative 1 close  and ( 3 , − 11 ) , open 3 comma negative 11 close comma  focus ( 3 , − 4 ) open 3 comma negative 4 close
10. vertices (9, 9) and ( 9 , − 5 ) , open 9 comma negative 5 close comma  focus (9, 6)
11. vertices ( − 5 , 4 ) open negative 5 comma 4 close  and (8, 4), focus ( − 4 , 4 ) open negative 4 comma 4 close

See Problem 2.

Identify the center, vertices, and foci of each hyperbola.

12. ( x + 11 ) 2 16 − y 2 9 = 1 fraction open , x plus 11 , close squared , over 16 end fraction . minus , fraction y squared , over 9 end fraction , equals 1
13. ( y − 4 ) 2 9 − ( x − 3 ) 2 4 = 1 fraction open , y minus 4 , close squared , over 9 end fraction . minus . fraction open , x minus 3 , close squared , over 4 end fraction . equals 1
14. ( y + 8 ) 2 4 − ( x + 3 ) 2 49 = 1 fraction open , y plus 8 , close squared , over 4 end fraction . minus . fraction open , x plus 3 , close squared , over 49 end fraction . equals 1

See Problem 3.

Identify each conic section by writing the equation in standard form and sketching the graph. For a parabola, give the vertex. For a circle, give the center and the radius. For an ellipse or a hyperbola, give the center and the foci.

15. x 2 − 8 x − y + 19 = 0 x squared , minus 8 x minus y plus 19 equals 0
16. 3 x 2 + 6 x + y 2 − 6 y = − 3 3 x squared , plus 6 x plus , y squared , minus 6 y equals negative 3
17. y 2 − x 2 + 6 x − 4 y = 6 y squared , minus , x squared , plus 6 x minus 4 y equals 6
18. x 2 − 4 y 2 − 2 x − 8 y = 7 x squared , minus , 4 y squared , minus 2 x minus 8 y equals 7
19. y 2 − 2 x − 4 y = − 10 y squared , minus 2 x minus 4 y equals negative 10
20. x 2 + y 2 − 4 x − 6 y − 3 = 0 x squared , plus , y squared , minus 4 x minus 6 y minus 3 equals 0

See Problem 4.

21. Navigation A lighthouse is on an island 4 miles from a long, straight shoreline. When a boat is directly between the lighthouse and the shoreline, it is 1 mile from the lighthouse and 3 miles from the shore. As it sails away from the shore and lighthouse, it continues so that the difference in distances between boat and lighthouse and between boat and shore is always 2 miles.
    1. What conic section models this problem?
    2. What part of the graph does the lighthouse represent? The shoreline?
    3. What equation represents the path of the boat?

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