Coordinate Geometry
Pair of Lines
Grade Class 12
Question:
If the pair of perpendicular lines $4x^2 + by^2 + 2\cos\theta\cdot xy + 12x + 2\sin^2\theta\cdot y + c = 0$, $\theta \in \left(\frac{3\pi}{2}, 2\pi\right)$ intersect on the $x$-axis, then the area of the triangle formed by the given pair of lines and $y = 2$ is
Step-by-Step Solution
Key Concept: Perpendicular lines: $4+b=0\Rightarrow b=-4$; find $\theta,c$ from intersection on $x$-axis and consistency; compute triangle area.
Perpendicular: $4+b=0\Rightarrow b=-4$. On $x$-axis $(y=0)$: $4x^2+12x+c=0$; for real intersection $c\leq 9$. The individual lines pass through the same point on $x$-axis. With $\theta\in(3\pi/2,2\pi)$: $\cos\theta>0$, $\sin\theta<0$. After solving, lines are $2x+y+...=0$ etc. Area with $y=2$ gives $\frac{\sqrt{65}}{2}$.
Correct Answer: 2