1. Let the centre of the circle be on the positive $x$-axis

Since the centre lies on the positive $x$-axis, let it be $O(h,0), \quad h>0.$ Let the radius be $r$.

2. Use the tangency condition with the line $x-y+1=0$

The distance of $O(h,0)$ from the line $x-y+1=0$ must equal the radius: $r=\frac{|h-0+1|}{\sqrt{1^2+(-1)^2}}=\frac{h+1}{\sqrt{2}}$ (since $h>0$, $h+1>0$).

So, r=\frac{h+1}{\sqrt{2}}. \tag{1}

3. Use the chord-length condition for the line $-3x+2y=1$

Rewrite the line as $-3x+2y-1=0.$

Distance of the centre $O(h,0)$ from this line is $d=\frac{|-3h+0-1|}{\sqrt{(-3)^2+2^2}}=\frac{3h+1}{\sqrt{13}}$ because $h>0$ implies $-3h-1<0$.

If a line at distance $d$ from the centre cuts a chord of length $L$, then $L=2\sqrt{r^2-d^2}.$ Given $L=\frac{4}{\sqrt{13}},$ so $2\sqrt{r^2-d^2}=\frac{4}{\sqrt{13}}$ $\sqrt{r^2-d^2}=\frac{2}{\sqrt{13}}$ r^2-d^2=\frac{4}{13}. \tag{2}

Now substitute $r$ and $d$: $\frac{(h+1)^2}{2}-\frac{(3h+1)^2}{13}=\frac{4}{13}.$ Multiply by $26$: $13(h+1)^2-2(3h+1)^2=8.$ Expand: $13(h^2+2h+1)-2(9h^2+6h+1)=8$ $13h^2+26h+13-18h^2-12h-2=8$ $-5h^2+14h+11=8$ $-5h^2+14h+3=0$ $5h^2-14h-3=0.$

Solve: $h=\frac{14\pm\sqrt{196+60}}{10}=\frac{14\pm16}{10}.$ Thus, $h=3 \quad \text{or} \quad h=-\frac15.$ Since the centre is on the positive $x$-axis, $h=3$.

Hence the centre of the circle is $O=(3,0).$

From (1), $r=\frac{3+1}{\sqrt2}=2\sqrt2.$

4. Form the hyperbola

Given hyperbola: $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1.$

For this standard hyperbola:

• centre is at the origin,
• foci are at $(\pm c,0)$ where $c^2=\alpha^2+\beta^2,$
• length of transverse axis is $2\alpha$.

We are told:

• one focus is the centre of the circle, i.e. $(3,0)$, so $c=3;$
• transverse axis length equals the diameter of the circle.

Diameter of the circle is $2r=4\sqrt2.$ So, $2\alpha=4\sqrt2 \implies \alpha=2\sqrt2 \implies \alpha^2=8.$

Now use $c^2=\alpha^2+\beta^2$ $9=8+\beta^2$ $\beta^2=1.$

5. Compute the required value

$2\alpha^2+3\beta^2=2(8)+3(1)=16+3=19.$

6. Comparison with stored answer

Derived answer = $19$. Stored correct answer = $19$.

They match.