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  • A straight line L with negative slope passes through the point (8, 2) and cuts the positive coordinate axes at points P and Q. Find the absolute minimum value of <img alt="\mathrm{OP + OQ}" src="ht

A straight line L with negative slope passes through the point (8, 2) and cuts the positive coordinate axes at points P and Q. Find the absolute minimum value of \mathrm{OP + OQ}, as L varies, where O is the origin.

Option: 1

16


Option: 2

18


Option: 3

12


Option: 4

14


Answers (1)

best_answer

Let the equation of the line be \mathrm{(y - 2) = m(x - 8) }where \mathrm{m < 0 }

\Rightarrow \mathrm{P \equiv\left(8-\frac{2}{m}, 0\right) \text { and } Q \equiv(0,2-8 m)}

Now  \mathrm{O P+O Q=\left|8-\frac{2}{m}\right|+|2-8 m|=10+\frac{2}{-m}+8(-m) \geq 10+2 \sqrt{\frac{2}{-m} \times 8(-m)} \geq 18}

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Gaurav

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