Straight Lines
Intercepts and perpendicular from origin
nta_pyq_2023_jan
Grade 11
Question:
A straight line cuts off the intercepts $OA = a$ and $OB = b$ on the positive directions of x-axis and y-axis respectively. If the perpendicular from origin O to this line makes an angle of $\dfrac{\pi}{6}$ with positive direction of y-axis and the area of $\triangle OAB$ is $\dfrac{98}{3}\sqrt{3}$, then $a^2 - b^2$ is equal to:
Step-by-Step Solution
Key Concept: The perpendicular from O makes angle $\pi/6$ with y-axis, i.e., $\pi/3$ with x-axis. The line equation: $\frac{x}{a} + \frac{y}{b} = 1$ in normal form is $x\cos\frac{\pi}{3} + y\sin\frac{\pi}{3} = p$. So $a = 2p$, $b = \frac{2p}{\sqrt{3}}$.
$p^2 = 49$, $a = 14$, $b = \frac{14}{\sqrt{3}}$. $a^2 - b^2 = 196 - \frac{196}{3} = \frac{392}{3}$.
Correct Answer: 1