Key Concepts and Formulas

• Distance from a point to a line: The perpendicular distance $d$ from a point $(x_0, y_0)$ to a line $Ax+By+C=0$ is given by $d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$.
• Chord Length and Distance: If a circle of radius $R$ has a chord of length $L$, and the perpendicular distance from the center of the circle to the chord is $d$, then $R^2 = d^2 + (\frac{L}{2})^2$.
• Hyperbola Properties: For a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the foci are at $(\pm ae, 0)$, the length of the transverse axis is $2a$, and $b^2 = a^2(e^2 - 1)$ or $a^2e^2 = a^2 + b^2$.

Step-by-Step Solution

Step 1: Define the Circle's Center and Radius

Since the center of the circle $C$ lies on the positive $x$-axis, let its coordinates be $(h, 0)$, where $h > 0$. Let the radius of the circle be $R$.

Step 2: Use the Tangency Condition

The circle touches the line $x - y + 1 = 0$. Therefore, the distance from the center $(h, 0)$ to this line is equal to the radius $R$. $R = \frac{|h - 0 + 1|}{\sqrt{1^2 + (-1)^2}} = \frac{|h + 1|}{\sqrt{2}}$ Since $h > 0$, $h + 1 > 0$, so $|h + 1| = h + 1$. Thus, $R = \frac{h + 1}{\sqrt{2}}$ $R^2 = \frac{(h + 1)^2}{2} \quad \text{.... (Equation 1)}$

Step 3: Use the Chord Length Information

The circle cuts off a chord of length $L = \frac{4}{\sqrt{13}}$ along the line $-3x + 2y = 1$, which can be written as $-3x + 2y - 1 = 0$. The distance $d$ from the center $(h, 0)$ to this line is: $d = \frac{|-3h + 2(0) - 1|}{\sqrt{(-3)^2 + 2^2}} = \frac{|-3h - 1|}{\sqrt{13}} = \frac{|3h + 1|}{\sqrt{13}}$ Since $h > 0$, $3h + 1 > 0$, so $|3h + 1| = 3h + 1$. Thus, $d = \frac{3h + 1}{\sqrt{13}}$ We know that $R^2 = d^2 + (\frac{L}{2})^2$. Plugging in the values, we get: $R^2 = \left(\frac{3h + 1}{\sqrt{13}}\right)^2 + \left(\frac{4}{2\sqrt{13}}\right)^2 = \frac{(3h + 1)^2}{13} + \frac{4}{13}$ $R^2 = \frac{9h^2 + 6h + 1 + 4}{13} = \frac{9h^2 + 6h + 5}{13} \quad \text{.... (Equation 2)}$

Step 4: Solve for $h$ and $R$

Equate Equation 1 and Equation 2: $\frac{(h + 1)^2}{2} = \frac{9h^2 + 6h + 5}{13}$ $13(h^2 + 2h + 1) = 2(9h^2 + 6h + 5)$ $13h^2 + 26h + 13 = 18h^2 + 12h + 10$ $0 = 5h^2 - 14h - 3$ $0 = (5h + 1)(h - 3)$ Since $h > 0$, we have $h = 3$. Now, substitute $h = 3$ into Equation 1: $R^2 = \frac{(3 + 1)^2}{2} = \frac{16}{2} = 8$ So, $R = \sqrt{8} = 2\sqrt{2}$.

Step 5: Find the Hyperbola's Properties

The center of the circle $C$ is $(3, 0)$, which is one of the foci of the hyperbola. Thus, $ae = 3$. The length of the transverse axis is equal to the diameter of the circle, so $2a = 2R = 4\sqrt{2}$, which means $a = 2\sqrt{2}$. Then $a^2 = (2\sqrt{2})^2 = 8$. Since $ae = 3$, we have $a^2e^2 = 9$. We also know $a^2e^2 = a^2 + b^2$. Therefore, $9 = 8 + b^2$, so $b^2 = 1$. Thus, $\alpha^2 = a^2 = 8$ and $\beta^2 = b^2 = 1$.

Step 6: Calculate $2\alpha^2 + 3\beta^2$

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

Common Mistakes & Tips

• Be careful with signs when calculating the distance from a point to a line. Use absolute values correctly.
• Remember the relationships between $a, b,$ and $e$ for a hyperbola.
• Double-check your algebraic manipulations to avoid errors.

Summary

We first found the center and radius of the circle using the tangency and chord length conditions. Then, using the information about the hyperbola's focus and transverse axis, we determined the values of $a^2$ and $b^2$. Finally, we calculated $2\alpha^2 + 3\beta^2$, which equals 19. However, the "Correct Answer" is given as 1. Let's re-evaluate.

Step 5 (Revised): Find the Hyperbola's Properties

The center of the circle $C$ is $(3, 0)$, which is one of the foci of the hyperbola. Thus, $ae = 3$. The length of the transverse axis is equal to the diameter of the circle, so $2a = 2R = 4\sqrt{2}$, which means $a = 2\sqrt{2}$. Then $a^2 = (2\sqrt{2})^2 = 8$. So $\alpha^2 = 8$. Since $ae = 3$, we have $e = \frac{3}{a} = \frac{3}{2\sqrt{2}}$. Then $e^2 = \frac{9}{8}$. We also know $b^2 = a^2(e^2 - 1)$. Therefore, $b^2 = 8(\frac{9}{8} - 1) = 8(\frac{1}{8}) = 1$. Thus, $\beta^2 = 1$.

Step 6 (Revised): Calculate $2\alpha^2 + 3\beta^2$ $2\alpha^2 + 3\beta^2 = 2(8) + 3(1) = 16 + 3 = 19$.

Still not getting 1 as the answer. Let us re-examine the equations.

$R = \frac{h+1}{\sqrt{2}}$ so $R^2 = \frac{(h+1)^2}{2}$ $d = \frac{3h+1}{\sqrt{13}}$ $R^2 = d^2 + (\frac{2}{\sqrt{13}})^2 = \frac{(3h+1)^2}{13} + \frac{4}{13}$ $\frac{(h+1)^2}{2} = \frac{(3h+1)^2+4}{13}$ $13(h^2+2h+1) = 2(9h^2+6h+1+4) = 2(9h^2+6h+5)$ $13h^2+26h+13 = 18h^2+12h+10$ $5h^2 - 14h - 3 = 0$ $(5h+1)(h-3)=0$ $h = 3$. $R = \frac{3+1}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2}$ $2a = 2R$, so $a = R = 2\sqrt{2}$. $a^2 = 8$. $ae = 3$ so $e = \frac{3}{2\sqrt{2}} = \frac{3\sqrt{2}}{4}$. $e^2 = \frac{9}{8}$. $b^2 = a^2(e^2-1) = 8(\frac{9}{8}-1) = 8(\frac{1}{8}) = 1$. $b^2 = 1$. Then $2a^2 + 3b^2 = 2(8)+3(1) = 19$.

There MUST be a mistake in the given answer. I have checked all the steps and the logic. Assuming the correct answer is 19.

The final answer is \boxed{19}.