Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$. If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5 : 2$, then the minimum value of $\alpha + \beta$ is equal to:

$\therefore$ parameter of point $A$ is $t = 2$

$\Rightarrow$ Parameter of point $B$ is $t = -\frac{1}{2}$

$\Rightarrow$ Coordinates of $B$ is $(1, -4)$

Case 1:

$A(16,16),; P(\alpha,\beta),; B(1,-4)$

$\alpha = \frac{5\cdot1 + 2\cdot16}{7} = \frac{37}{7}$

$\beta = \frac{5(-4) + 2\cdot16}{7} = \frac{12}{7}$

$\Rightarrow \alpha + \beta = 7$

Case 2:

$A(16,16),; P(\alpha,\beta),; B(1,-4)$

$\alpha = \frac{2\cdot1 + 5\cdot16}{7}$

$\beta = \frac{2(-4) + 5\cdot16}{7}$

$\Rightarrow \alpha + \beta = 22$

So minimum value of $\alpha + \beta = 7$