Consider an ellipse with semiaxes a and b (a>b) and a circle of radius b, the ce
ID: 2968807 • Letter: C
Question
Consider an ellipse with semiaxes a and b (a>b) and a circle of radius b, the center of the circle lying on the extension of the major axis of the ellipse. Show that for every line parallel to major axis of the ellipse, the portion of that line inside the ellipse will be a/b times the portion inside the circle. Use this fact and Cavalieri's principle to compute the area of the ellipse.
Cavalieri's principe implies that figures in a plane lying between two parallel lines and such that all sections parallel to those lines have the same length must have equal area.
Consider an ellipse with semiaxes a and b (a>b) and a circle of radius b, the center of the circle lying on the extension of the major axis of the ellipse. Show that for every line parallel to major axis of the ellipse, the portion of that line inside the ellipse will be a/b times the portion inside the circle. Use this fact and Cavalieri's principle to compute the area of the ellipse. Cavalieri's principe implies that figures in a plane lying between two parallel lines and such that all sections parallel to those lines have the same length must have equal area.Explanation / Answer
Consider the standard ellipse x^2/a^2+y^2/b^2=1 and a circle of radius b with centre on x axis.
Consider a parallel line to major axis, at a distance y from the major axis. Then the corresponding length on the circle is 2*root(b^2-y^2).
If it intersects the ellipse at (x,y) then x^2/a^2 + y^2/b^2 =1 , and the length on the ellipse is given by 2x.
The ratio = x/root(b^2-y^2) = x/(bx/a) = a/b.
From the Cavalieri's principle, Area of ellipse = a/b * pi*b^2= pi*a*b.