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JEE Main 20246 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
6 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Straight Lines

JEE Main 2024 · 6 April · Morning Shift · Mathematics · Straight Lines

Let a variable line of slope m>0 passing through the point (4,-9)...

Let a variable line of slope m>0m>0 passing through the point (4,−9)(4,-9) intersect the coordinate axes at the points AA and BB. The minimum value of the sum of the distances of AA and BB from the origin is

Options

  1. A30
  2. B25
  3. C15
  4. D10

Correct answer

B. 25

Step-by-step solution

equation of line is y+9=m(x−4)∴A=(9+4mm,0)B=(0,−9−4m)∴OA+OB=9+4mm+9+4m\begin{aligned} & y+9=m(x-4) \\ & \therefore \quad A=\left(\frac{9+4 m}{m}, 0\right) \\ & \quad B=(0,-9-4 m) \\ & \therefore \quad O A+O B=\frac{9+4 m}{m}+9+4 m \end{aligned}∵m>0=13+9 m+4 m∵4 m+9 m2≥36⇒4 m+9 m≥12∴OA+OB≥25\begin{aligned} & \because \mathrm{m}>0 \\ & =13+\frac{9}{\mathrm{~m}}+4 \mathrm{~m} \\ & \because \frac{4 \mathrm{~m}+\frac{9}{\mathrm{~m}}}{2} \geq \sqrt{36} \Rightarrow 4 \mathrm{~m}+\frac{9}{\mathrm{~m}} \geq 12 \\ & \therefore \mathrm{OA}+\mathrm{OB} \geq 25\end{aligned}$
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