document.write( "Question 1045317: how can i find out the equation for ellipse when i have foci and length of major axis is given? \n" ); document.write( "
Algebra.Com's Answer #660735 by KMST(5328)\"\" \"About 
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You also need to know the coordinates of the center, and
\n" ); document.write( "what direction the major axis follows.
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\n" ); document.write( "An ellipse is a stretched circle.
\n" ); document.write( "A circle of radius \"R\" with center at the origin has the equation
\n" ); document.write( "\"x%5E2%2By%5E2=R%5E2\" <---> \"x%5E2%2FR%5E2%2By%5E2%2FR%5E2=1\" ,
\n" ); document.write( "because \"x%5E2%2By%5E2\" is the square of the distance from point (x,y) to origin (0,0) .
\n" ); document.write( "Thankfully, we almost always get confronted by ellipses that are easier to calculate,
\n" ); document.write( "where the longest axis of symmetry, the major axis, is either
\n" ); document.write( "parallel to the x-axis (and we call it horizontal), or
\n" ); document.write( "parallel to the y-axis (and we call it vertical).
\n" ); document.write( "An ellipse with center at the origin and axes of symmetry parallel to the x- and y-axes has the equation
\n" ); document.write( "\"x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1\" or \"x%5E2%2Fb%5E2%2By%5E2%2Fa%5E2=1\" , with \"a%3Eb\" .
\n" ); document.write( "If the center of a circle, or ellipse is not \"O%280%2C0%29\",
\n" ); document.write( "but a point \"C%28h%2Ck%29\" ,
\n" ); document.write( "you just write \"%28x-k%29%5E2\" instead of \"x%5E2\" ,and
\n" ); document.write( "\"%28y-k%29%5E2\" instead of \"y%5E2\" .
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\n" ); document.write( "The distance \"c\" from each focus to the center, is related to \"a\" and \"b\" by \"a%5E2=b%5E2%2Bc%5E2\" , so you only need to know two of those distances.
\n" ); document.write( "Given foci and length of major axis, you have \"a\" and \"c\" ,
\n" ); document.write( "just find \"b%5E2=a%5E2-c%5E2\" and plug the found \"b%5E2\" value into the equation.
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\n" ); document.write( "The points the farthest from the center are the vertices,
\n" ); document.write( "located on the major axis, at a distance \"a\" from the center.
\n" ); document.write( "The points the closest to the center are often called co-vertices,
\n" ); document.write( "located on the minor axis, at a distance \"b\" from the center.
\n" ); document.write( "The foci are located on the major axis, at a distance \"c\" to either side of the center.
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\n" ); document.write( "Here is an ellipse. you see the origin, point \"O%280%2C0%29\" .
\n" ); document.write( "You see one labeled vertex, point \"A%28a%2C0%29%7D%7D+%2C+at+a+distance+%7B%7B%7Ba\" from the center.
\n" ); document.write( "I labeled the foci, focus \"F%28c%2C0%29\" , and Focus \"E%28-c%2C0%29\" ,
\n" ); document.write( "both at a distance \"c\" from the center.
\n" ); document.write( "I also labeled one of the co-vertices, point \"B%280%2Cb%29\" , at a distance \"b\" from the center.
\n" ); document.write( "The fancy definition of ellipse says that for all the points on the ellipse,
\n" ); document.write( "the sum of the distances to one Focus and the other is the same.
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\n" ); document.write( "You can see that point \"a\" is at a distance \"a-c\" from focus \"F\" ,
\n" ); document.write( "and at a distance (\"a%2Bc\" from focus \"E\".
\n" ); document.write( "So the sum of the distance to the foci is
\n" ); document.write( "\"a=c%2Ba%2Bc=2a\" for point \"A\", and for all points in the ellipse.
\n" ); document.write( "Then, point \"B\" is at the same distance \"a\" from focus \"E\" and focus \"F\" .
\n" ); document.write( "Applying the Pythagorean theorem to right triangle \"FBO\" ,
\n" ); document.write( "you get the relationship \"a%5E2=b%5E2%2Bc%5E2\" . \n" ); document.write( "
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