Question

Let $E_1$ and $E_2$ be two ellipses whose centers are at origin. The major axes of $E_1$ and $E_2$ lie along the $x-$axis and $y-$axis, respectively. Let $S$ be the circle $x^2+(y-1{)}^2=2.$ The straight line $x+y=3$ touches the curves $S,E_1$ and $E_2$ at $P,Q$ and $R$, respectively. Suppose that $PQ=PR=\frac{2\sqrt{2}}{3}$. If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$, respectively then the correct expression(s) is (are)

• A. $e_1^2+e_2^2=\frac{43}{40}$
• B. $e_1{e}_2=\frac{\sqrt{7}}{2\sqrt{10}}$
• C. $|e_1^2-e_2^2|=\frac{5}{8}$
• D. $e_1{e}_2=\frac{\sqrt{3}}{4}$

Answer 1

Answer: (A), (B)

The correct options are
A $e_1^2+e_2^2=\frac{43}{40}$
B $e_1{e}_2=\frac{\sqrt{7}}{2\sqrt{10}}$

For given line point of contact for $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $\frac{a^2}{3},\frac{b^2}{3}$

And for $E_2:\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$ is $\frac{A^2}{3},\frac{B^2}{3}$

Point of contact of $x+y=3$ and circle is $(1,2)$.

General point on $x+y=3$ is $(1\mp \frac{r}{\sqrt{2}},2\pm \frac{r}{\sqrt{2}})$.

So, required points are $(\frac{1}{3},\frac{8}{3})$ and $(\frac{5}{3},\frac{4}{3})$.

Comparing with the points of contact of ellipse $a^2=5,b^2=4,B^2=8,A^2=1$

$e_1{e}_2=\frac{\sqrt{7}}{2\sqrt{10}}$

${e_1}^2+{e_2}^2=\frac{43}{40}$

$e_1=\sqrt{1-\frac{4}{5}}=\frac{1}{\sqrt{5}}$

$e_2=\sqrt{1-\frac{1}{8}}=\frac{\sqrt{7}}{2\sqrt{2}}$