Let $E_1$ and $E_2$ be two ellipse whose centers are at the origin. The major axes of $E_{1}and E_2$ lie along the x - axis and the y - axis, respectively. Let S be the circle $x^2+{\left(y-1\right)}^2=2.$ 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)

• Correct answer
  A

  $e_1^2+ e_2^2=\frac{43}{40}$
• Correct answer
  B

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

  $\left|e_1^2- e_2^2\right|=\frac{5}{8}$
• D

  $e_1{e}_2= \frac{\sqrt{3}}{4}$

Hence, the correct answer is options: 'A', 'B'.