Prove the formula for the foci of an ellipseI have summarized the question below:If the vertices of an ellipse centered at the origin are (a,0),(-a,0),(0,b), and (0,-b), and a &gt; b, prove that for foci at ( ± c , 0 ), c 2 = a 2 − b 2 .I am guessing that I have to use the distance formula, which is d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .

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Prove the formula for the foci of an ellipseI have summarized the question below:If the vertices of an ellipse centered at the origin are (a,0),(-a,0),(0,b), and (0,-b), and   a  &amp;gt;  b, prove that for foci at   (  ±  c  ,  0  ),       c    2    =      a    2    −      b    2  .I am guessing that I have to use the distance formula, which is   d  =      (          x      2        −          x      1              )      2        +    (          y      2        −          y      1              )      2      .

Lance Long · Accepted Answer

Step 1Any point on the ellipse is such that   M      F    1    +  M      F    2    =  A      F    1    +  A      F    2    =  2  a where       F    1    ,      F    2   are the foci and A is the (a,0) vertex. So let&#039;s write that for B(0,b)            c      2        +          b      2        +            c      2        +          b      2        =  2  aStep 2This rewrites easily as       c    2    +      b    2    =      a    2

Braylon Lester · Accepted Answer

Step 1With drawing ellipse shape with center at the origin and are   A  (  ±  a  ,  0  ) and   B  (  0  ,  ±  b  ) are vertices, find a symmetric shape and symmetric foci at F(c,0) and F′(-c,0). Step 2With definition for ellipse for exery point X on ellipse       |    X  F      |    +      |    X      F    ′        |    =  2  a so       |    B  F      |    +      |    B      F    ′        |    =  2  a so   2            b      2        +          c      2        =  2  a that concludes       c    2    =      a    2    −      b    2  .

If the vertices of an ellipse centered at the origin are (a, 0), (-a, 0), (0, b), and (0, -b), and a>b, prove that for foci at (pm c, 0), c^2=a^2-b^2.

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