- #1
Mary89
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Find the equation of a ellipse given the foci. (1,0) (3,4)
The general equation of an ellipse in standard form is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse, and a and b are the lengths of the semi-major and semi-minor axes, respectively.
The center of the ellipse can be found by taking the midpoint between the foci, which in this case would be ((1+3)/2, (0+4)/2) = (2,2). This point will be the (h,k) values in the general equation of the ellipse.
Yes, the eccentricity of an ellipse can be found by the formula e = c/a, where c is the distance between the foci and a is the length of the semi-major axis. In this case, the distance between the foci is √((3-1)^2 + (4-0)^2) = √20, and the semi-major axis is (3-1)/2 = 1. Therefore, the eccentricity is √20/1 = √20.
The lengths of the semi-major and semi-minor axes can be found by using the distance formula. The length of the semi-major axis is half the distance between the foci, so in this case, it would be √((3-1)^2 + (4-0)^2)/2 = √10. The length of the semi-minor axis can be found by using the Pythagorean theorem, so it would be √(a^2 - c^2) = √(10-5) = √5.
To graph the ellipse, you can use the center point and the lengths of the semi-major and semi-minor axes to plot the four vertices of the ellipse. Then, use a compass to draw the ellipse by setting the compass to the length of the semi-major axis and drawing arcs from each vertex. Finally, connect the arcs to create the ellipse.