1. **Problem statement:** Find the locus of a point $P(\alpha, \beta)$ such that the line $y = \alpha x + \beta$ is tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1.$$\n\n2. **Formula and approach:** The condition for tangency of a line $y = mx + c$ to a conic is that the quadratic equation formed by substituting $y = mx + c$ into the conic has a discriminant equal to zero. Here, $m = \alpha$ and $c = \beta$.\n\n3. **Substitute $y = \alpha x + \beta$ into the hyperbola:**\n$$\frac{x^2}{a^2} - \frac{(\alpha x + \beta)^2}{b^2} = 1.$$\nMultiply both sides by $a^2 b^2$ to clear denominators:\n$$b^2 x^2 - a^2 (\alpha x + \beta)^2 = a^2 b^2.$$\n\n4. **Expand and rearrange:**\n$$b^2 x^2 - a^2 (\alpha^2 x^2 + 2 \alpha \beta x + \beta^2) = a^2 b^2,$$\nwhich simplifies to\n$$b^2 x^2 - a^2 \alpha^2 x^2 - 2 a^2 \alpha \beta x - a^2 \beta^2 = a^2 b^2.$$\nGroup terms by powers of $x$:\n$$\left(b^2 - a^2 \alpha^2\right) x^2 - 2 a^2 \alpha \beta x - a^2 \beta^2 - a^2 b^2 = 0.$$\n\n5. **Quadratic in $x$:**\n$$A x^2 + B x + C = 0,$$\nwhere\n$$A = b^2 - a^2 \alpha^2,$$\n$$B = -2 a^2 \alpha \beta,$$\n$$C = -a^2 \beta^2 - a^2 b^2.$$\n\n6. **Tangency condition:** The discriminant $\Delta$ must be zero:\n$$\Delta = B^2 - 4 A C = 0.$$\nCalculate:\n$$B^2 = 4 a^4 \alpha^2 \beta^2,$$\n$$4 A C = 4 (b^2 - a^2 \alpha^2)(-a^2 \beta^2 - a^2 b^2).$$\n\n7. **Set discriminant to zero:**\n$$4 a^4 \alpha^2 \beta^2 - 4 (b^2 - a^2 \alpha^2)(-a^2 \beta^2 - a^2 b^2) = 0.$$\nDivide both sides by 4:\n$$a^4 \alpha^2 \beta^2 = (b^2 - a^2 \alpha^2)(-a^2 \beta^2 - a^2 b^2).$$\n\n8. **Expand right side:**\n$$= (b^2 - a^2 \alpha^2)(-a^2 (\beta^2 + b^2)) = -a^2 (b^2 - a^2 \alpha^2)(\beta^2 + b^2).$$\n\n9. **Rewrite equation:**\n$$a^4 \alpha^2 \beta^2 = -a^2 (b^2 - a^2 \alpha^2)(\beta^2 + b^2).$$\nDivide both sides by $a^2$:\n$$a^2 \alpha^2 \beta^2 = - (b^2 - a^2 \alpha^2)(\beta^2 + b^2).$$\n\n10. **Expand right side:**\n$$- (b^2 - a^2 \alpha^2)(\beta^2 + b^2) = - b^2 \beta^2 - b^4 + a^2 \alpha^2 \beta^2 + a^2 \alpha^2 b^2.$$\n\n11. **Bring all terms to one side:**\n$$a^2 \alpha^2 \beta^2 + b^2 \beta^2 + b^4 - a^2 \alpha^2 \beta^2 - a^2 \alpha^2 b^2 = 0,$$\nwhich simplifies to\n$$b^2 \beta^2 + b^4 - a^2 \alpha^2 b^2 = 0.$$\n\n12. **Divide by $b^2$ (assuming $b \neq 0$):**\n$$\beta^2 + b^2 - a^2 \alpha^2 = 0,$$\nor\n$$a^2 \alpha^2 - \beta^2 = b^2.$$\n\n13. **Final locus equation:**\n$$\boxed{a^2 \alpha^2 - \beta^2 = b^2}.$$\n\nThis is the equation of the locus of the point $P(\alpha, \beta)$ such that the line $y = \alpha x + \beta$ is tangent to the given hyperbola.