A circle with centre $$(2, 3)$$ and radius $$4$$ intersects the line $$x + y = 3$$ at the points $$P$$ and $$Q$$. If the tangents at $$P$$ and $$Q$$ intersect at the point $$S(\alpha, \beta)$$, then $$4\alpha - 7\beta$$ is equal to

Given circle with center $$(2, 3)$$ and radius $$4$$ is $$(x - 2)^2 + (y - 3)^2 = 16$$. The line $$x + y = 3$$ is the chord of contact from the point of intersection of the tangents, $$S(\alpha, \beta)$$.

The equation of the chord of contact from $$S(\alpha, \beta)$$ is given by $$T = 0$$:

$$(\alpha - 2)(x - 2) + (\beta - 3)(y - 3) = 16$$

$$(\alpha - 2)x + (\beta - 3)y - (2\alpha + 3\beta + 3) = 0$$

Comparing the coefficients of this line with the given line $$x + y - 3 = 0$$:

$$\frac{\alpha - 2}{1} = \frac{\beta - 3}{1} = \frac{2\alpha + 3\beta + 3}{3}$$

From the first two ratios:

$$\alpha - 2 = \beta - 3 \implies \beta = \alpha + 1$$

From the first and third ratios:

$$3(\alpha - 2) = 2\alpha + 3\beta + 3 \implies \alpha - 3\beta = 9$$

Substituting $$\beta = \alpha + 1$$ into the linear relation:

$$\alpha - 3(\alpha + 1) = 9 \implies -2\alpha = 12 \implies \alpha = -6$$

$$\beta = -6 + 1 = -5$$

Evaluating the required expression for $$4\alpha - 7\beta$$:

$$4\alpha - 7\beta = 4(-6) - 7(-5) = -24 + 35 = 11$$

Conclusion:

The value of $$4\alpha - 7\beta$$ is equal to 11.