1. Given ellipse

   $\frac{x^2}{9}+\frac{y^2}{4}=1$

   Here, $a^2=9,\quad b^2=4$

2. Chord with given midpoint

   Midpoint of the chord is $(x_1,y_1)=\left(\sqrt{2},\frac{4}{3}\right)$

   For the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ the chord whose midpoint is $(x_1,y_1)$ is given by the midpoint form:

   $\frac{xx_1}{a^2}+\frac{yy_1}{b^2}=\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}$

   Substituting values:

   $\frac{x\sqrt{2}}{9}+\frac{y(4/3)}{4}=\frac{(\sqrt{2})^2}{9}+\frac{(4/3)^2}{4}$

   $\frac{\sqrt{2}x}{9}+\frac{y}{3}=\frac{2}{9}+\frac{16/9}{4}$

   $\frac{\sqrt{2}x}{9}+\frac{y}{3}=\frac{2}{9}+\frac{4}{9}=\frac{6}{9}=\frac{2}{3}$

   Multiply by $9$:

   $\sqrt{2}x+3y=6$

3. Find the points of intersection of this chord with the ellipse

   From the line,

   $3y=6-\sqrt{2}x \implies y=2-\frac{\sqrt{2}}{3}x$

   Substitute into ellipse:

   $\frac{x^2}{9}+\frac{1}{4}\left(2-\frac{\sqrt{2}}{3}x\right)^2=1$

   Expand:

   $\left(2-\frac{\sqrt{2}}{3}x\right)^2=4-\frac{4\sqrt{2}}{3}x+\frac{2}{9}x^2$

   So,

   $\frac{x^2}{9}+\frac{1}{4}\left(4-\frac{4\sqrt{2}}{3}x+\frac{2}{9}x^2\right)=1$

   $\frac{x^2}{9}+1-\frac{\sqrt{2}}{3}x+\frac{1}{18}x^2=1$

   $\frac{x^2}{9}+\frac{x^2}{18}-\frac{\sqrt{2}}{3}x=0$

   $\frac{x^2}{6}-\frac{\sqrt{2}}{3}x=0$

   $x\left(\frac{x}{6}-\frac{\sqrt{2}}{3}\right)=0$

   Hence,

   $x=0 \quad \text{or} \quad x=2\sqrt{2}$

   Corresponding $y$ values:

   • If $x=0$, then $y=2$
   • If $x=2\sqrt{2}$, then $y=2-\frac{\sqrt{2}}{3}(2\sqrt{2})=2-\frac{4}{3}=\frac{2}{3}$

   So the endpoints of the chord are:

   $(0,2),\quad (2\sqrt{2},\tfrac{2}{3})$

4. Find the length of the chord

   $L=\sqrt{(2\sqrt{2}-0)^2+\left(\frac{2}{3}-2\right)^2}$

   $=\sqrt{8+\left(-\frac{4}{3}\right)^2}$

   $=\sqrt{8+\frac{16}{9}}$

   $=\sqrt{\frac{72+16}{9}}=\sqrt{\frac{88}{9}}=\frac{2\sqrt{22}}{3}$

5. Compare with given form

   Given chord length is

   $\frac{2\sqrt{\alpha}}{3}$

   Therefore,

   $\frac{2\sqrt{\alpha}}{3}=\frac{2\sqrt{22}}{3}$

   So,

   $\alpha=22$

6. Option check

   • A: $26$ ❌
   • B: $18$ ❌
   • C: $22$ ✅
   • D: $20$ ❌

Thus, the correct answer is Option C.