10-3 Circles

Quick Review

In a plane, a circle is the set of all points that are a given distance, the radius, r, from a given point, the center, (h, k).

Example

Write an equation in standard form of a circle with center ( − 3 , 4 ) open negative 3 comma 4 close  and radius 2.

Use the standard form of the equation of a circle. Substitute − 3 negative 3  for h, 4 for k, and 2 for r.

( x − h ) 2 + ( y − k ) 2 = r 2 open x minus h close squared . plus . open y minus k close squared . equals , r squared

( x − ( − 3 ) ) 2 + ( y − 4 ) 2 = 2 2 open x minus open negative 3 close close squared . plus . open y minus 4 close squared . equals , 2 squared

( x + 3 ) 2 + ( y − 4 ) 2 = 4 open x plus 3 close squared . plus . open y minus 4 close squared . equals 4

An equation for the circle is ( x + 3 ) 2 + ( y − 4 ) 2 = 4 . open x plus 3 close squared . plus . open y minus 4 close squared . equals 4 .

Exercises

Write an equation in standard form of a circle with the given center and radius.

22. center (0, 0); radius 4
23. center (8, 1); radius 5

Write an equation for each translation of x 2 + y 2 = r 2 x squared , plus , y squared , equals , r squared  with the given radius.

24. left 3 units, up 2 units; radius 10
25. right 5 units, down 3 units; radius 8

Find the center and the radius of each circle. Graph each circle. Describe the translation from center (0, 0).

26. ( x − 1 ) 2 + y 2 = 64 open x minus 1 close squared . plus , y squared , equals 64
27. ( x + 7 ) 2 + ( y + 3 ) 2 = 49 open x plus 7 close squared . plus . open y plus 3 close squared . equals 49

10-4 Ellipses

Quick Review

An ellipse is the set of all points P, where the sum of the distances between P and two fixed points, the foci, is constant. The major axis contains the foci, and its endpoints are the vertices of the ellipse. For a > b , eh greater than b comma  there are two standard forms of ellipses centered at the origin. If x 2 a 2 + y 2 b 2 = 1 , fraction x squared , over eh squared end fraction . plus . fraction y squared , over b squared end fraction . equals 1 comma  the major axis is horizontal with vertices ( ± a , 0 ) , open plus minus eh comma 0 close comma  foci ( ± c , 0 ) , open plus minus c comma 0 close comma  and co-vertices ( 0 , ± b ) . open 0 comma plus minus b close .  If x 2 b 2 + y 2 a 2 = 1 , fraction x squared , over b squared end fraction . plus . fraction y squared , over eh squared end fraction . equals 1 comma  the major axis is vertical with vertices ( 0 , ± a ) , open 0 comma plus minus eh close comma  foci ( 0 , ± c ) , open 0 comma plus minus c close comma  and co-vertices ( ± b , 0 ) . open plus minus b comma 0 close .

In either case, c 2 = a 2 − b 2 . c squared , equals , eh squared , minus , b squared , .

Example

Write an equation of an ellipse with foci ( ± 5 , 0 ) open plus minus 5 comma 0 close  and co-vertices ( 0 , ± 3 ) . open 0 comma plus minus 3 close .

Since the foci are ( ± 5 , 0 ) , open plus minus 5 comma 0 close comma  the major axis is horizontal. Since c = 5 c equals 5  and b = 3 , c 2 = 25 b equals 3 comma , c squared , equals 25  and b 2 = 9 . b squared , equals 9 .  Using the equation c 2 = a 2 − b 2 , a 2 = 34 . c squared , equals , eh squared , minus , b squared , comma . eh squared , equals 34 .

An equation of the ellipse is x 2 34 + y 2 9 = 1 . fraction x squared , over 34 end fraction , plus , fraction y squared , over 9 end fraction , equals 1 .

Exercises

Write an equation of an ellipse centered at the origin, satisfying the given conditions.

28. foci ( ± 1 , 0 ) ; open plus minus 1 comma 0 close semicolon  co-vertices ( 0 , ± 4 ) open 0 comma plus minus 4 close
29. vertex ( 0 , 29 ) ; open 0 comma square root of 29 close semicolon  co-vertex ( − 5 , 0 ) open negative 5 comma 0 close
30. focus (0, 1); vertex ( 0 , 10 ) open 0 comma square root of 10 close
31. foci ( ± 2 , 0 ) ; open plus minus 2 comma 0 close semicolon  co-vertices ( 0 , ± 6 ) open 0 comma plus minus 6 close
32. Write an equation of an ellipse centered at the origin with height 8 units and width 16 units.
33. Find the foci of the graph of x 2 4 + y 2 9 = 1 . fraction x squared , over 4 end fraction , plus , fraction y squared , over 9 end fraction , equals 1 .  Graph the ellipse.

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