Algebra 2
Algebra 2
1st Edition
ISBN: 9780078884825
Author: McGraw-Hill/Glencoe
Publisher: Glencoe/McGraw-Hill School Pub Co
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Vertices Of An Ellipse
In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…
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Chapter 10.4, Problem 31PPS
To determine

To find: The coordinates of the centre and foci and the length of the major and minor axis for the given ellipse.

Expert Solution & Answer
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Answer to Problem 31PPS

The coordinates of the centre and foci are (4,−3) and 1 (4,−3±26) , the length of the major axis is 3 47 unit , and the length of the minor axis is 4 unit.

Explanation of Solution

Given information:

The equation of the ellipse is,

  3x2+2y2+12x−20y+14=0

Concept used:

Write the standard form of the equation of the ellipse whose orientation is along the horizontal axis is,

  (x−h)2a2+(y−k)2b2=1

The coordinate of the foci is (h±c,k)

The value of c is calculated as,

  c=a2−b2

The vertices are at (h±a,k) and the co-vertices are at (h,k±b).

Write the standard form of the equation of the ellipse whose orientation is along the vertical axis is,

  (y−k)2a2+(x−h)2b2=1

The coordinate of the foci is (h,k±c)

The vertices are at (h,k±a) and the co-vertices are at (h±b,k).

The value of c is calculated as,

  c=a2−b2

The equation of the ellipse can be written as,

  3x2+2y2+12x−20y+14=03(x2+4x+4)−12+2(y2−10y+25)−50+14=03(x+2)2+2(y−5)2=48(x+2)216+(y−5)224=1

Write the standard form of the equation of the ellipse whose orientation is along the vertical axis is,

  (y−k)2a2+(x−h)2b2=1

Compare the given equation with the standard form of the equation to obtain,

  h=−2k=5a2=24a=26

  b2=16b=4

The length of the major axis is calculated as,

  2a=2×262a=46 unit

The length of the minor axis is calculated as,

  2b=2×42b=8 unit

The value of c is calculated as,

  c=a2−b2=24−16=22 unit

The coordinate of the foci is calculated as,

  (h,k±c)=(−2,5±22)

Consider the following graph of the ellipse as,

  Algebra 2, Chapter 10.4, Problem 31PPS

Thus, the coordinates of the centre and foci are (−2,5) and (−2,5±22) , the length of the major axis is 46 unit , and the length of the minor axis is 8 unit.

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Chapter 10.4, Problem 30PPS
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Chapter 10.4, Problem 32PPS

Chapter 10 Solutions

Algebra 2

Ch. 10.1 - Prob. 1ACYPCh. 10.1 - Prob. 1BCYPCh. 10.1 - Prob. 2CYPCh. 10.1 - Prob. 3CYPCh. 10.1 - Prob. 1CYUCh. 10.1 - Prob. 2CYUCh. 10.1 - Prob. 3CYUCh. 10.1 - Prob. 4CYUCh. 10.1 - Prob. 5CYUCh. 10.1 - Prob. 6CYU
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