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

### Related Formula textFor a focal chord with ends (at_1^2, 2at_1) text and (at_2^2, 2at_2), text the relation is t_1t_2 = -1$$\text{For a focal chord with ends } (at_1^2, 2at_1) \text{ and } (at_2^2, 2at_2), \text{ the relation is } t_1t_2 = -1$$ textSection formula: (x, y) = left( fracmx_2 + nx_1m+n, fracmy_2 + ny_1m+n right)$$\text{Section formula: } (x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)$$ ### Core Logic

For y^2 = 16x$y^2 = 16x$, a = 4$a = 4$. The given point A(16, 16)$A(16, 16)$ is equivalent to 4t^2 = 16$4t^2 = 16$ and 2(4)t = 16$2(4)t = 16$, which gives parameter t_1 = 2$t_1 = 2$. The other end B$B$ has parameter t_2 = -frac1t_1 = -frac12$t_2 = -\frac{1}{t_1} = -\frac{1}{2}$. ### Step 1: Calculate coordinates of B For t_2 = -1/2$t_2 = -1/2$, point B$B$ is: x = 4(-1/2)^2 = 1$x = 4(-1/2)^2 = 1$ y = 8(-1/2) = -4$y = 8(-1/2) = -4$ So, B(1, -4)$B(1, -4)$. ### Step 2: Section formula calculations (Two cases) Point P(alpha, beta)$P(\alpha, \beta)$ divides AB$AB$ in the ratio 5:2$5:2$. There are two possibilities depending on which end the ratio starts from. Case 1: Ratio 5 from B to A (i.e. A is x_2$x_2$ and B is x_1$x_1$): alpha = frac5(16) + 2(1)7 = frac80 + 27 = frac827$$\alpha = \frac{5(16) + 2(1)}{7} = \frac{80 + 2}{7} = \frac{82}{7}$$ beta = frac5(16) + 2(-4)7 = frac80 - 87 = frac727$$\beta = \frac{5(16) + 2(-4)}{7} = \frac{80 - 8}{7} = \frac{72}{7}$$ Sum: alpha + beta = frac1547 = 22$\alpha + \beta = \frac{154}{7} = 22$. Case 2: Ratio 5 from A to B (i.e. B is x_2$x_2$ and A is x_1$x_1$): alpha = frac5(1) + 2(16)7 = frac5 + 327 = frac377$$\alpha = \frac{5(1) + 2(16)}{7} = \frac{5 + 32}{7} = \frac{37}{7}$$ beta = frac5(-4) + 2(16)7 = frac-20 + 327 = frac127$$\beta = \frac{5(-4) + 2(16)}{7} = \frac{-20 + 32}{7} = \frac{12}{7}$$ Sum: alpha + beta = frac497 = 7$\alpha + \beta = \frac{49}{7} = 7$. ### Step 3: Minimum Value Comparing the two possible sums, 7 < 22$7 < 22$. Thus, the minimum value is 7$7$. ### Pattern Recognition When a line segment is divided in a given ratio, 'internal division' inherently bears two solutions based on the orientation (from point A or point B). Always evaluate both cases when finding a minimum or maximum. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Conic Sections