document.write( "Question 694400: Hey I was hoping you could help me out. My math problem is the following.
\n" ); document.write( "9x^2+25^2=1. I understand that this is the formula for an ellipse, but I am unaware of how to proceed with finding the vertices, the foci, and how to graph it. Am I supposed to divide both side by 9 and 25 to achieve x^2/a+y^2/b=1? My problem is making the right side equal out to 1. Thanks for any help! \n" ); document.write( "

Algebra.Com's Answer #427956 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
You probably meant
\n" ); document.write( " $\"9x%5E2%2B25y%5E2=1\"$
\n" ); document.write( "and want to express it in the form
\n" ); document.write( " $\"x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1\"$ ,
\n" ); document.write( "which is appropriate for an ellipse centered at the origin, with axes along the x- and y-axes.
\n" ); document.write( "
\n" ); document.write( "You could write your equation as
\n" ); document.write( " $\"x%5E2%2F%281%2F9%29%2By%5E2%2F%281%2F25%29=1\"$ or as $\"x%5E2%2F%281%2F3%29%5E2%2By%5E2%2F%281%2F5%29%5E2=1\"$
\n" ); document.write( "
\n" ); document.write( "Even without re-writing the equation, it was obvious that it was the equation for an ellipse centered at the origin, with axes along the x- and y-axes, because there were no terms in $\"x\"$ or $\"y\"$ , or $\"xy\"$ .
\n" ); document.write( "
\n" ); document.write( "VERTICES:
\n" ); document.write( "The vertices, then, are the intersections with the x- and y-axes, were $\"y=0\"$ and $\"x=0\"$ , and those are points you need to graph the ellipse.
\n" ); document.write( " $\"y=0\"$ --> $\"9x%5E2=1\"$ --> $\"x%5E2=1%2F9\"$ --> $\"x+=1%2F3\"$ or $\"x=-1%2F3\"$ gives you vertices (-1/3,0) and (1/3,0).
\n" ); document.write( " $\"x=0\"$ --> $\"25y%5E2=1\"$ --> $\"y%5E2=1%2F25\"$ --> $\"y=1%2F5\"$ or $\"x=-1%2F5\"$ gives you co-vertices (0,-1/5) and (0,1/5).
\n" ); document.write( "
\n" ); document.write( "AXES:
\n" ); document.write( "Vertices (-1/3,0) and (1/3,0) are farther from center (0,0) than co-vertices (0,-1/5) and (0,1/5):
\n" ); document.write( " $\"1%2F3%3E1%2F5\"$ ,
\n" ); document.write( "so the segment between (-1/3,0) and (1/3,0) is called the major axis,
\n" ); document.write( "and the segment/distance from each of those vertices to the center is called the semi-major axis, represented as $\"a\"$ ,
\n" ); document.write( "so $\"a=1%2F3\"$ .
\n" ); document.write( "The distance from the center to the co-vertices is the semi-minor axis:
\n" ); document.write( " $\"b=1%2F5\"$ .
\n" ); document.write( "
\n" ); document.write( "FOCI:
\n" ); document.write( "The foci would be useful to draw the ellipse if you wanted a very accurate representation.
\n" ); document.write( "They are points on both semi-major axes at a distance from the center called the focal distance, represented as $\"c\"$ .
\n" ); document.write( "Since for all points of the ellipse the sum of the distances to the foci is constant, the same.
\n" ); document.write( "For the vertices, the distance to the nearest focus is $\"a-c\"$ and the distance to the other one is $\"a%2Bc\"$ , so the sum is $\"2a\"$ .
\n" ); document.write( "For the co-vertices, the distance to each focus in the hypotenuse of a right triangle with leg lengths $\"b\"$ and $\"c\"$ .
\n" ); document.write( "Each of those distances is $\"sqrt%28b%5E2%2Bc%5E2%29\"$ , so the sum is $\"2sqrt%28b%5E2%2Bc%5E2%29\"$ .
\n" ); document.write( "Since that should be the same as for the vertices,
\n" ); document.write( " $\"2a=2sqrt%28b%5E2%2Bc%5E2%29\"$ <--> $\"a=sqrt%28b%5E2%2Bc%5E2%29\"$ --> $\"highlight%28a%5E2=b%5E2%2Bc%5E2%29\"$
\n" ); document.write( "That formula allows you to find the focal distance and graph the foci.
\n" ); document.write( " $\"%281%2F3%29%5E2=%281%2F5%29%5E2%2Bc%5E2\"$ --> $\"1%2F9=1%2F25%2Bc%5E2\"$ --> $\"c%5E2=1%2F9-1%2F25\"$ --> $\"c%5E2=%2825-9%29%2F225\"$ --> $\"c%5E2=16%2F225\"$ --> $\"highlight%28c=4%2F15%29\"$
\n" ); document.write( "So the foci will be at (-4/15,0) and (4/15,0).
\n" ); document.write( "
\n" ); document.write( "GRAPHING:
\n" ); document.write( "We can plot the vertices co-vertices and foci, like this
\n" ); document.write( "

Then draw the ellipse $\"graph%28300%2C300%2C-0.4%2C0.4%2C-0.4%2C0.4%2C9x%5E2%2B25y%5E2%3C1%29\"$
\n" ); document.write( "I would just draw a curve that passes through vertices and co-vertices and looks like it could be an ellipse.
\n" ); document.write( "To have a real ellipse, you would have to stick pins at the foci; make a loop of thread with a total length (all around) equal to the the distance between the foci, plus the distance between the vertices; throw the loop over the pins; stretch the loop with the tip of a pencil, and draw around keeping the loop fully stretched. \n" ); document.write( "

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