Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
The vertices $B$ and $C$ of a triangle $ABC$ lie on the lines $3y = 4x$ and $y = 0$ respectively and the side $BC$ passes through the point $\left(\frac{2}{3}, \frac{1}{3}\right)$. If $ABOC$ is a rhombus, $O$ being the origin and the coordinates of $A$ are $(h, k)$, then $\frac{h}{k}$ is equal to ______.
Step-by-Step Solution
Key Concept: Using trigonometric ratios to express point coordinates and substituting into line equations determines unknown parameters.
Given $\angle BOC = \theta$ with $\tan \theta = \frac{4}{3}$, we have $\sin \theta = \frac{4}{5}$ and $\cos \theta = \frac{3}{5}$. With $OC = t$, the coordinates of $B$ are $(\frac{3t}{5}, \frac{4t}{5})$ and $C$ is at $(t, 0)$. The line $BC$ has equation $y = -2(x-t)$, and this passes through $(\frac{2}{3}, \frac{2}{3})$ giving $t = 1$. The coordinates of $A$ are $h = \frac{8}{5}$ and $k = 2$.
Correct Answer: 2