SOLUTION: in an ellipse why does c squared = a squared minus b squared

Question 958578: in an ellipse why does c squared = a squared minus b squared
Answer by Edwin McCravy(20060) (Show Source): You can put this solution on YOUR website!



The vertices are Q and U,  P is a covertex.
The foci (focal points) are R and T.
The center is S.

SQ = SU = a
SP = b
SR = ST = c

By using the Pythagorean theorem on right triangle PST, we have

ST� + PS� = PT�

c� + b� = PT�

     c� = PT� - b�

If we can show that PT = a we will have shown what we needed to.

By definition, every ellipse is such that any two points on the ellipse are such
that the sum of the two distances from one of them to the two focal points
(foci) equals the sum of the two distances from the other point to the two focal
points (foci).

P and U are two points on the ellipse.  So by the definition of an ellipse,
PR + PT = UR + UT

Since PR = PT and UT = QR,

PT + PT = UR + UT

    2PT = UR + QR

    2PT = QU

Since SQ = SU = a, then QU = SQ + SU = a + a = 2a

   2PT = 2a

    PT = a

That was what we needed to show, since we have shown that

   c� = PT� - b�.

So now we can substitute "a" for PT, and have what we wanted
to show:

   c� = a� - b�

Edwin

RELATED QUESTIONS
Starting from the definition of an ellipse, can you show me how you get x squared/a... (answered by Fombitz)

a/b=c/d then value of... (answered by ewatrrr)

What is 64a squared minus 25b squared c... (answered by Alan3354)

b squared + x squared = a... (answered by Alan3354)

Show (a+b)*squared* is not equal to a*squared* + b*squared*... (answered by Alan3354)

18 squared minus 98... (answered by Fombitz)

Factor a squared minus ab minus 20b... (answered by ankor@dixie-net.com)

Is the Fourmula for pathogram therom is A squared + b squared = C... (answered by JBarnum,richwmiller)

a(squared)+n +2b(squared)= (a+b)(a+2b) What does n represent? A.ab B.2ab C.3ab (answered by oscargut)