If the normal at the points where the straight line $lx + my = 1$ meet the parabola $y^2 = 4ux$, meet at the point $(h, k)$ on the parabola $y^2 = 4ux$, then $\frac{kl}{am}$ is equal to____.
Step-by-Step Solution
Key Concept: The normals at two points on a parabola where a line intersects it meet at a point, and this constraint combined with the parabola equation yields a fixed ratio relationship.
Let the line $lx + my = 1$ intersect the parabola $y^2 = 4ax$ at points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$. The normal at a point $(at^2, 2at)$ on the parabola is $y = -tx + 2at + at^3$. Since both normals pass through $(h,k)$, we have $k = -t_1h + 2at_1 + at_1^3$ and $k = -t_2h + 2at_2 + at_2^3$, which means $t_1$ and $t_2$ satisfy $t^3 + (2a-h)t/a - k/a = 0$. From the line equation: $lat_i^2 + m(2at_i) = 1$ gives $lat^2 + 2mat - 1 = 0$. Using Vieta's formulas on both cubic and quadratic, and the condition that $(h,k)$ lies on the parabola $k^2 = 4ah$, we can establish that $ rac{kl}{m} = 4a$.
Correct Answer: 4