Determine the equation of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ such that it has the least area but contains the circle $(x-1)^2+y^2=1$
Since the area of ellipse is $A=\pi ab\Rightarrow A^2=\pi^2a^2b^2$ and the circle and the ellipse touch each other internally. This much I can visualise, but how to find out $a$ and $b$ from this?
Best Answer
Firstly, define $f(x):=b\sqrt{1-\frac{x^2}{a^2}}$ and $g(x):=\sqrt{1-(x-1)^2}$ such that $f$ and $g$ represent the ellipse and the circle respectively in the upper half of the coordinate system.
In order to guarantee that the ellipse contains the circle, we need to have: $$ f(x)≥g(x)\iff b\sqrt{1-\frac{x^2}{a^2}}≥\sqrt{1-(x-1)^2} \iff b^2-\frac{b^2}{a^2}x^2≥2x-x^2\iff \\ \left(1-\frac{b^2}{a^2}\right)x^2-2x+b^2≥0 $$ In the last inequality, we have a quadratic function, which needs to be nonnegative everywhere, thus the discriminant $D$ has to be nonpositive: $$ D=4-4\left(1-\frac{b^2}{a^2}\right)b^2≤0\iff 1≤b^2-\frac{b^4}{a^2} \iff \frac{b^4}{a^2}≤b^2-1 $$ We can see that $0<b^2-1\iff 1<b$, therefore we can divide by $b^2-1$ without any changes: $$ a^2≥\frac{b^4}{b^2-1}\iff a≥\frac{b^2}{\sqrt{b^2-1}} $$ This last inequality is sufficient and necessary for the given condition.
Therefore, we have: $$ A=\pi ab≥\frac{\pi b^3}{\sqrt{b^2-1}} $$ Thus, the minimum of $A$ has to be greater than or equal to the minimum of $\frac{\pi b^3}{\sqrt{b^2-1}}$. This minimum can be found as follows:
Define $h(b):=\frac{\pi b^3}{\sqrt{b^2-1}}$. $$ h'(b)=\frac{\pi b^2\left(2b^2-3\right)}{\left(b^2-1\right)^{\frac{3}{2}}} \implies \left(h'(b)=0\iff b=\sqrt{\frac{3}{2}}\right) $$ $b=0$ is impossible because $b>1$. Therefore, the minimum has to be at $b=\sqrt{\frac{3}{2}}$ so we have $\min A≥h\left(\sqrt{\frac{3}{2}}\right)=\frac{\pi 3\sqrt{3}}{2}$. This minimum can indeed be achieved for $a=\frac{3}{\sqrt 2}$ and $b=\sqrt{\frac{3}{2}}$, for which we also have: $$ a=\frac{3}{\sqrt 2}≥\frac{\sqrt{\frac{3}{2}}^2}{\sqrt{\sqrt{\frac{3}{2}}^2-1}}=\frac{b^2}{\sqrt{b^2-1}} $$ Thus, the equation of the ellipse has to be: $$ \frac{2x^2}{9}+\frac{2y^2}{3}=1 $$