Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
If $L = \left(\frac{1}{x_l}\right), M = \left(\frac{1}{x_m}\right), N = \left(\frac{1}{x_n}\right)$ where $x_k \neq 0$, denotes the $k^{th}$ terms of a H.P. for $k \in N$, then:
$ar(\Delta LMN) = \frac{l^2m^2n^2}{2}\left[(l-m)^2 + (m-n)^2 + (n-l)^2\right]$
$\Delta LMN$ is a right angled triangle
The points $L, M, N$ are collinear
$\Delta LMN$ is equilateral
Step-by-Step Solution
Key Concept: Reciprocals of intercepts in AP implies the three points lie on the same straight line.
For a straight line, the reciprocals of intercepts form an arithmetic progression. If $\frac{1}{x_a} = a + (l-1)d$ and $\frac{1}{x_n} = a + (n-1)d$, then these values satisfy the given AP condition. The determinant condition $\begin{vmatrix} a+(l-1)d & l & 1 \\ a+(m-1)d & m & 1 \\ a+(n-1)d & n & 1 \end{vmatrix} = 0$ confirms collinearity of the three points on the line.
Correct Answer: 3