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JEE Main 20249 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
9 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Application of Derivatives

JEE Main 2024 · 9 April · Morning Shift · Mathematics · Application of Derivatives

A variable line L passes through the point (3,5) and intersects the...

A variable line LL passes through the point (3,5)(3,5) and intersects the positive coordinate axes at the points A\mathrm{A} and B\mathrm{B}. The minimum area of the triangle OAB\mathrm{OAB}, where O\mathrm{O} is the origin, is :A variable line L passes through the point (3,5) and intersects the positive coordinate axes at the points A and B .

Options

  1. A30
  2. B25
  3. C40
  4. D35

Correct answer

A. 30

Step-by-step solution

xa+yb=13a+5b=1⇒b=5aa−3,a>3\begin{aligned} & \frac{x}{a}+\frac{y}{b}=1 \\ & \frac{3}{a}+\frac{5}{b}=1 \Rightarrow b=\frac{5 a}{a-3}, a>3\end{aligned} A variable line L passes through the point (3,5) and intersects the positive coordinate axes at the points A and B .
A=12ab=12a5a(a−3)=52⋅a2a−3A=\frac{1}{2} a b=\frac{1}{2} a \frac{5 a}{(a-3)}=\frac{5}{2} \cdot \frac{a^2}{a-3}=52(a2−9+9a−3)=52(a+3+9a−3)=52(a−3+9a−3+6)≥30\begin{aligned} & =\frac{5}{2}\left(\frac{\mathrm{a}^2-9+9}{\mathrm{a}-3}\right) \\ & =\frac{5}{2}\left(\mathrm{a}+3+\frac{9}{\mathrm{a}-3}\right) \\ & =\frac{5}{2}\left(\mathrm{a}-3+\frac{9}{\mathrm{a}-3}+6\right) \geq 30\end{aligned}$$
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