Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
Let $A\left(\frac{1}{2}, 0\right), B\left(\frac{3}{2}, 0\right), C\left(\frac{5}{2}, 0\right)$ be the given points and $P$ be a point satisfying $\max(PA + PB, PB + PC) < 2$. All points $P$ are points common to:
two ellipse
two hyperbola
a circle and an ellipse
a circle and a hyperbola
Step-by-Step Solution
Key Concept: The locus of points satisfying sum-of-distances conditions lies within an ellipse, and area is computed via integration.
For the ellipse with foci A and B where $PA + PB < 2$ and $PB + PC < 2$ (with C being another focus), the locus of P is the interior region symmetric about the x-axis. Using $2a = \frac{3}{2} - \frac{1}{2} = 1$, we get $b^2 = \frac{3}{4}$. The area of the shaded region is computed as $4\int_0^{1/\sqrt{2}} \frac{\sqrt{3}}{2}\sqrt{1-(x-1)^2}\,dx = \sqrt{3}(\frac{\pi}{3} - \frac{\sqrt{3}}{4})$.
Correct Answer: 1