by Som Phene
Last Updated August 10, 2018 04:20 AM

Consider an ellipse with major and minor axes of length 10 and 8 resp. The radius of the largest circle that can be inscribed in this ellipse, given that the centre of this circle is one of the focus of the ellipse is.

i attempted to solve this by using a very simple concept: all points of the circle should either lie inside or on the circle. hence assuming any point on the circle to be x=ae+rcos(theta) and y=rsin(theta), should satisfy : x^2/a^2+y^2/b^2-1<0 i got a quadratic in cos(theta) and i made the equation to be true independent of theta which gave r lies in [a-ae,a+ae] what is wrong to solve it by this method. I would also like to know any other methods to solve this problem.

Hint:

- Parametrize the ellipse as $(x,y) = (5 \cos t, 4 \sin t)$.
- Note that a focus is located at $(3,0)$.
- Obtain an expression for the distance of a point on the ellipse to this focus.
- Find a value of $t$ for which this distance is minimized.
- Compute the minimum distance. This is the radius of the largest circle centered at the focus.

Let the ellipse be x^2/a^2 + y^2/b^2 =1. Parametric point on the ellipse is (acost, bsint) . Distance of centre (0,0) from this point is given by (a-aecost). (Substitute in distance formula and simplify to obtain that expression)

(a-aecost) in this case is (5-3cost).

It is minimum for cost=1 and its value is 2. Hence , the radius of the largest circle that can be inscribed is 2 units.

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