Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
It is desired to construct a right angled triangle $ABC$ ($\angle C = \pi/2$) in xy-plane so that its sides are parallel to co-ordinates axes and the medians through $A$ and $B$ lie on the lines $y = 3x+1$ and $y = mx+2$ respectively. The values of $m$ for which such a triangle is possible is/are:
Step-by-Step Solution
Key Concept: The angle bisector divides the angle into two parts whose tangents are related by the geometric configuration; case analysis handles different orderings of slopes.
Using the angle bisector property, $\tan\theta_1 = \frac{BC}{2AC}$ and $\tan\theta_2 = \frac{2BC}{AC}$, giving $\frac{\tan\theta_1}{\tan\theta_2} = \frac{1}{4}$. For **Case I** ($m 3$): $\tan\theta_1 = \frac{1}{m}$ and $\tan\theta_2 = \frac{1}{3}$ yields $m = 12$.
Correct Answer: 2