Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
Let $(3,4)$ be a fixed point. A straight line passing through this point cuts the positive direction of the coordinate axes at the points $P$ and $Q$. If $\lambda$ sq unit the minimum area of the triangle $OPQ$, $O$ being the origin. Then the value of $\lambda$ must be ____.
Step-by-Step Solution
Key Concept: Area of a triangle formed by a line and coordinate axes can be minimized using AM-GM inequality on the slope terms.
A line through $(3,4)$ with slope $m$ is $y - 4 = m(x-3)$. The area of triangle $AOB$ formed with axes intercepts is $A = \frac{1}{2}|24 + (-9m) - \frac{16}{m}|$. By AM-GM inequality on the variable terms, $\frac{-9m - \frac{16}{m}}{2} \geq \sqrt{9 \times 16} = 12$, giving minimum area $A_{\min} = 24$ when $m = -\frac{4}{3}$.
Correct Answer: 24