Let the point mathrmP$\mathrm{P}$ of the focal chord mathrmPQ$\mathrm{PQ}$ of the parabola mathrmy^2 = 16mathrmx$\mathrm{y}^2 = 16\mathrm{x}$ be (1, -4)$(1, -4)$. If the focus of the parabola divides the chord mathrmPQ$\mathrm{PQ}$ in the ratio mathrmm : mathrmn$\mathrm{m} : \mathrm{n}$, gcd(mathrmm, mathrmn) = 1$\gcd(\mathrm{m}, \mathrm{n}) = 1$, then mathrmm^2 + mathrmn^2$\mathrm{m}^2 + \mathrm{n}^2$ is equal to:

### Related Formula textParametric coordinates on y^2 = 4ax: quad (at^2, \, 2at)$$\text{Parametric coordinates on } y^2 = 4ax: \quad (at^2, \, 2at)$$ textFocal Chord relation: t_1 t_2 = -1$$\text{Focal Chord relation: } t_1 t_2 = -1$$ textSection Formula: (x_c, y_c) = left( fracm x_2 + n x_1m+n, \, fracm y_2 + n y_1m+n right)$$\text{Section Formula: } (x_c, y_c) = \left( \frac{m x_2 + n x_1}{m+n}, \, \frac{m y_2 + n y_1}{m+n} \right)$$ ### Core Logic We find the parametric parameters of coordinates P$P$ and Q$Q$, obtain their Cartesian values, and then apply the section formula with the focus S$S$ to calculate the splitting ratio. ### Step 1: Find coordinates of P and Q For parabola y^2 = 16x$y^2 = 16x$, the focal parameter is a = 4$a = 4$. Focus is S(4, 0)$S(4, 0)$. Let P$P$ be (a t_1^2, 2a t_1) = (1, -4)$(a t_1^2, 2a t_1) = (1, -4)$: 2a t_1 = -4 implies 2(4) t_1 = -4 implies t_1 = -frac12$$2a t_1 = -4 \implies 2(4) t_1 = -4 \implies t_1 = -\frac{1}{2}$$ Since PQ$PQ$ is a focal chord, the parametric points are coupled: t_1 t_2 = -1 implies t_2 = 2$$t_1 t_2 = -1 \implies t_2 = 2$$ Now, calculate the coordinates of Q$Q$: Q equiv (a t_2^2, \, 2 a t_2) = (4(4), \, 2(4)(2)) = (16, \, 16)$$Q \equiv (a t_2^2, \, 2 a t_2) = (4(4), \, 2(4)(2)) = (16, \, 16)$$ ### Step 2: Solve for the dividing ratio Let the focus S(4, 0)$S(4, 0)$ divide the line segment PQ$PQ$ internally in the ratio lambda : 1$\lambda : 1$. Using the y$y$-coordinate of the section formula: y_s = fraclambda y_q + 1 y_plambda + 1$$y_s = \frac{\lambda y_q + 1 y_p}{\lambda + 1}$$ 0 = fraclambda(16) + 1(-4)lambda + 1 implies 16lambda - 4 = 0 implies lambda = frac14$$0 = \frac{\lambda(16) + 1(-4)}{\lambda + 1} \implies 16\lambda - 4 = 0 \implies \lambda = \frac{1}{4}$$ Thus, the focus S$S$ divides the chord internally in the ratio 1:4$1:4$. Since gcd(1, 4) = 1$\gcd(1, 4) = 1$, we have m = 1$m = 1$ and n = 4$n = 4$: m^2 + n^2 = 1^2 + 4^2 = 1 + 16 = 17$$m^2 + n^2 = 1^2 + 4^2 = 1 + 16 = 17$$ ### Pattern Recognition Harmonic Mean Shortcut: In any parabola, the focus divides a focal chord internally into segments of lengths SP$SP$ and SQ$SQ$ such that the semi-latus rectum 2a$2a$ is the harmonic mean of these segments: frac1SP + frac1SQ = frac1a$\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Conic Sections