Answer to Question #339891 in General Chemistry for nixx

Question

1.give the coordinates of the foci , vertices and covertices of the ellipse wih equation x²/169 + y²/25 = 1 sketch the graph and include these points

2.find the equation in standard form of the ellipse whose foci are F1(-8,0) and F2 (8,0) such that for any point on it, the sum of its distances from the foci is 20.

3. An ellipse has vertices (-10,-4) and (6,-4) and covertices (-3,-9) and (-2,1). Find its standard equation and its foci.

Answer 1

Simplify each term in the equation in order to set the right side equal to 1

1

. The standard form of an ellipse or hyperbola requires the right side of the equation be 1

1

.

x

2

169

+

y

2

25

=

1

x2169+y225=1

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

(

x

−

h

)

2

a

2

+

(

y

−

k

)

2

b

2

=

1

(x-h)2a2+(y-k)2b2=1

Match the values in this ellipse to those of the standard form. The variable a

a

 represents the radius of the major axis of the ellipse, b

b

 represents the radius of the minor axis of the ellipse, h

h

 represents the x-offset from the origin, and k

k

 represents the y-offset from the origin.

a

=

13

a=13

b

=

5

b=5

k

=

k=0

h

=

h=0

The center of an ellipse follows the form of (

h

,

k

)

(h,k)

. Substitute in the values of h

h

 and k

k

.

(

,

)

(0,0)

Find c

c

, the distance from the center to a focus.

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12

12

Find the vertices.

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Vertex

1

Vertex1

: (

13

,

)

(13,0)

Vertex

2

Vertex2

: (

−

13

,

)

(-13,0)

Find the foci.

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Focus

1

Focus1

: (

12

,

)

(12,0)

Focus

2

Focus2

: (

−

12

,

)

(-12,0)

Find the eccentricity.

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12

13

1213

These values represent the important values for graphing and analyzing an ellipse.

Center: (

,

)

(0,0)

Vertex

1

Vertex1

: (

13

,

)

(13,0)

Vertex

2

Vertex2: (−13,0)(-13,0)Focus

1Focus1: (12,0)(12,0)Focus2

Focus2: (−12,0)(-12,0)

Eccentricity: 12131213