Straight Lines
Straight Line
Allen Star Batch
Grade 11
Question:
Let $A(1,1)$ and $B(3,3)$ be two fixed points and $P$ be a variable point such that area of $\triangle PAB$ remains constant equal to 1 for all position of $P$, then locus of $P$ is given by:
$2y = 2x + 1$
$2y = 2x - 1$
$y = x + 1$
$y = x - 1$
Step-by-Step Solution
Key Concept: The locus of points equidistant from a line and a point forms a parabola, but here we find the directrix-like property.
The locus of point $P$ equidistant from a line and a point (focus) is a parabola. Here, $P$ is at perpendicular distance $\frac{1}{\sqrt{2}}$ from line $y = x$, and distance $2\sqrt{2}$ from point $B(3,3)$. The locus of $P$ is parallel to $y = x$ at distance $\frac{1}{\sqrt{2}}$, giving $y - x = 1$ or $y - x = -1$.
Correct Answer: 3,4