Question

Find the equation of the largest circle with centre $(1,0)$ that can be inscribed in the ellipse $x^2+4y^2=16$.

• A. $(x-1{)}^2+y^2=\frac{11}{3}$
• B. $(x-1{)}^2+(y-1{)}^2=\frac{11}{3}$
• C. $(x{)}^2+(y-1{)}^2=\frac{11}{3}$
• D. $(x-1{)}^2+y^2=\frac{-11}{3}$

Answer 1

The correct option is A $(x-1{)}^2+y^2=\frac{11}{3}$

Let the common point where ellipse and circle will touch be $(4cos\theta ,2sin\theta )$

The equation of normal to ellipse is $4xsec\theta -2ycosec\theta =12$

The normal equation will pass through $(1,0)$ because it is also normal to circle

So we get $sec\theta =3$

So the point is $(\frac{4}{3},\frac{4\sqrt{2}}{3})$

Let the circle equation be ${(x-1)}^2+y^2=k$

By substituting the point in the circle equation , we get $k=\frac{33}{9}=\frac{11}{3}$

Therefore option $A$ is correct