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.
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.