10-1 Exploring Conic Sections

Quick Review

A conic section is formed by the intersection of a plane and a double cone. Circles, ellipses, parabolas, and hyperbolas are all conic sections.

Example

Graph the equation x 2 + y 2 = 9 . x squared , plus . y squared , equals 9 .  Identify the conic section, the domain and range.

Plot points that satisfy the equation. Connect them with a smooth curve.

The graph is a circle with center (0, 0) and radius 3.

The domain is − 3 ≤ x ≤ 3 . negative 3 less than or equal to x less than or equal to 3 .

The range is − 3 ≤ y ≤ 3 . negative 3 less than or equal to y less than or equal to 3 .

Exercises

Graph each equation. Identify the conic section, any lines of symmetry, and the domain and range.

6. x 2 49 + y 2 121 = 1 fraction x squared , over 49 end fraction , plus . fraction y squared , over 121 end fraction . equals 1
7. x 2 + y 2 = 4 x squared , plus . y squared , equals 4
8. x 2 25 − y 2 4 = 1 fraction x squared , over 25 end fraction , minus , fraction y squared , over 4 end fraction , equals 1
9. x = 2 y 2 + 5 x equals , 2 y squared , plus 5

Identify the center and domain and range of each graph.

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10-2 Parabolas

Quick Review

In a plane, a parabola is the set of all points that are the same distance, c, from a fixed point, the focus and a fixed line, the directrix.

For y = a x 2 , y equals , eh x squared , comma  if a > 0 , eh greater than 0 comma  the parabola opens up, and has focus (0, c) and directrix y = − c ; y equals negative c semicolon  if a < 0 , eh less than 0 comma  the parabola opens down, and has focus ( 0 , − c ) open 0 comma negative c close  and directrix y = c . y equals c .

For x = a y 2 , x equals , eh y squared , comma  if a > 0 , eh greater than 0 comma  the parabola opens right, and has focus (c, 0) and directrix x = − c ; x equals negative c semicolon  if a < 0 , eh less than 0 comma  the parabola opens left, and has focus ( − c , 0 ) open negative c comma 0 close  and directrix x = c . x equals c .  In all cases, a = 1 4 c . eh equals , fraction 1 , over 4 c end fraction , .

Example

Write an equation of a parabola that opens up, with vertex at the origin and focus 1 unit from the vertex.

Since the parabola opens up, use y = a x 2 . y equals , eh x squared , .  Since the focus is 1 unit from the vertex, c = 1 . c equals 1 .

a = 1 4 c = 1 4 ( 1 ) = 1 4 eh equals , fraction 1 , over 4 c end fraction , equals . fraction 1 , over 4 , open 1 close end fraction . equals , 1 fourth

An equation for the parabola is y = 1 4 x 2 . y equals , 1 fourth , x squared , .

Exercises

Write an equation of a parabola with vertex at the origin and the given focus.

12. (5, 0)
13. ( 0 , − 5 ) open 0 comma negative 5 close
14. (0, 6)

Write an equation of a parabola that opens up, with vertex at the origin and a focus as described.

15. focus is 2.5 units from the vertex
16. focus is 1 12 1 twelfth  of a unit from the vertex

Write an equation of a parabola with the given focus and directrix.

17. focus: (0, 3); directrix: y = − 1 y equals negative 1
18. focus: ( − 2 , 0 ) ; open negative 2 comma 0 close semicolon  directrix: x = 4 x equals 4

Find the focus and the directrix of the graph of each equation. Sketch the graph.

19. y = 5 x 2 y equals , 5 x squared
20. x = 2 y 2 x equals , 2 y squared
21. x = − 1 8 y 2 x equals negative , 1 eighth , y squared

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