Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
Let $S$ be the focus of $y^2 = 4x$ and a point $P$ is moving on the curve such that its abscissa is increasing at the rate of 4 units/sec, then the rate of increase of projection of $SP$ on $x + y = 1$ when $P$ is at $(4, 4)$ is:
$-\sqrt{2}$
$-1$
$\sqrt{2}$
$-\frac{3}{\sqrt{2}}$
Step-by-Step Solution
Key Concept: Express projection length in terms of angle and use parametric differentiation on the parabola.
The projection of point $P(x,y)$ on line $x + y = 1$ is $L = SP\cos(3\pi/4 - \theta)$ where $\theta$ is the slope angle of $SP$. For point $P$ on $y^2 = 4x$, compute $dy/dt = 8/y$ and $d\theta/dt = -2/5$. At $x = y = 4$, we get $\sec^2\theta = 25/9$, leading to $dL/dt = -\sqrt{2}$.
Correct Answer: 1