Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
Two equal sides $OA$ and $OB$ of an isosceles triangle lie in the first quadrant. If the slopes of $OA$ and $OB$ are $\frac{7}{17}$ and $1$, respectively and the length of perpendicular from $O$ to $AB$ is $\sqrt{13}$, if the equation of the side $AB$ is $ax + by = c$ then the value of $(c - a - b)$ is __________.
Step-by-Step Solution
Key Concept: Perpendicularity conditions on two pairs of lines determine the relationships between slope parameters, which then constrain the line equation through coordinate ratios.
Given $BC \perp AD$ with $\frac{\mu - 2\delta}{4\mu - \delta} = 1$ and $AC \perp BE$ with $\frac{2\delta + \lambda}{\delta - \lambda} = -4$, we derive $\mu - 2\delta = 4\mu - \delta \Rightarrow 3\mu = -\delta$ and $2\delta = \lambda$. Using these relations with the coordinate conditions $\frac{x}{y} = \frac{6-4+3}{-1-6+6} = -5$, we obtain $x + 5y = 0$ as the equation of line $AB$.
Correct Answer: 8