Knowing that the vertices of the ellipse are (-6,0) and (6,0) and the foci are (-sqrt(2231)/10, 0) and (sqrt(2231)/10, 0) . Find the ratio between the area of the circle and the ellipse.
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To find the area ratio between the circle and the ellipse, we need to find the equation of the ellipse and the equation of the circle first.
The center of the ellipse is the midpoint between the vertices, which is ((-6 + 6)/2, (0 + 0)/2) = (0, 0).
The distance between the center of the ellipse and one of the foci is sqrt( (-sqrt(2231)/10)^2 + 0^2) = sqrt(2231)/10.
Therefore, the distance between the center and one of the vertices is sqrt(2231)/10 * 2 = sqrt(2231)/5.
Since the vertices lie on the major axis, the semi-major axis of the ellipse is sqrt(2231)/5.
The equation of the 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.
Plugging in the values for the center and semi-major axis, we get:
(x-0)^2/( (sqrt(2231)/5)^2 ) + (y-0)^2/b^2 = 1,
x^2/( (sqrt(2231)/5)^2 ) + y^2/b^2 = 1,
x^2/( 2231/25 ) + y^2/b^2 = 1,
25x^2/2231 + y^2/b^2 = 1.
The center of the circle is also (0, 0). Since the foci of the ellipse lie on the x-axis:
The radius of the circle is the distance between the center and one of the foci, which is sqrt( (-sqrt(2231)/10)^2 + 0^2 ) = sqrt(2231)/10.
The equation of the circle in standard form is:
(x-h)^2 + (y-k)^2 = r^2,
(x-0)^2 + (y-0)^2 = (sqrt(2231)/10)^2,
x^2 + y^2 = 2231/100.
Now we can find the ratio between the area of the circle and the area of the ellipse:
Area of the ellipse = π * a * b,
Area of the circle = π * r^2,
Area ratio = (π * r^2) / (π * a * b),
Area ratio = r^2 / (a * b).
Plugging in the values for r, a, and b, we get:
Area ratio = ( (sqrt(2231)/10)^2 ) / ( (sqrt(2231)/5) * b ),
Area ratio = 2231/100 / ( (sqrt(2231)/5) * b ),
Area ratio = 2231 / 100 * (5/sqrt(2231)) * (1/b ),
Area ratio = 1 / ( 2 * sqrt(2231) * b ).
Since the definition of an ellipse is the set of all points where the sum of the distance to the foci is constant, we know that b > sqrt(2231)/10. Thus, b = a * sqrt(1 - e^2), where e is the eccentricity of the ellipse. Since the foci are on the x-axis, e = c/a, where c is the distance between one of the foci and the center (sqrt(2231)/10).
Therefore, b = a * sqrt(1 - (sqrt(2231)/10)^2), b = a * sqrt(1 - 1/100), b = a * sqrt(99/100), b = a * sqrt(99)/10.
Plugging in the value of b into the area ratio equation, we get:
Area ratio = 1 / (2 * sqrt(2231) * (a * sqrt(99)/10) ),
Area ratio = 1 / ( 2 * 10 * sqrt(2231) * sqrt(99) * a ),
Area ratio = 1 / ( 20 * sqrt(2231 * 99) * a ),
Area ratio = 1 / ( 20 * sqrt(220869) * a ),
Area ratio = 1 / ( 20 * 469 * a ),
Area ratio = 1 / ( 9380 * a ).
Therefore, the ratio between the area of the circle and the ellipse is 1 : (9380 * a).
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