Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
In the adjacent figure $ABC$ is right angled at $B$. If $AB = 4$ and $BC = 3$ and side $AC$ slides along the coordinate axes in such a way that $B$ always remains in the first quadrant, then $B$ always lie on straight line:
$y = x$
$3y = 4x$
$3x = 4y$
$3y + 4x = 0$
Step-by-Step Solution
Key Concept: The complementary angle $\frac{\pi}{2} - \theta$ relates $\tan\theta$ to the perpendicular line's slope via the cotangent function.
Given $\tan\theta = \frac{3}{4}$, the slope of line $OB$ is $\tan(\frac{\pi}{2} - \theta) = \cot\theta$. Since $\tan\theta = \frac{3}{4}$, we have $\cot\theta = \frac{4}{3}$. The locus of point $(h,k)$ on line $OB$ with slope $\frac{4}{3}$ is $k = \frac{4}{3}h$, which simplifies to $3y - 4x = 0$.
Correct Answer: 2