Straight Line and Pair of Straight Lines 1 Question 52
52. 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 $O P+O Q$, as $L$ varies, where $O$ is the origin. $\quad(2002,5 M)$
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Solution:
- Let $L:(y-2)=m(x-8), m<0$
The ponts $P$ and $Q$ are $8-\frac{2}{m}, 0$ and $(0,2-8 m)$, respectively.
Then, $O P+O Q=8-\frac{2}{m}+(2-8 m)=10+-\frac{2}{m}+(-8 m)$
[using $AM \geq GM$ ]
$\Rightarrow \frac{2}{-m}+(-8 m) \geq 2 \sqrt{16} \quad\left[\because \frac{2}{m}\right.$ and $-8 m$ are positive $]$
$\Rightarrow \quad-\frac{2}{m}+8 m \geq 8$
$\Rightarrow \quad 10-\frac{2}{m}+8 m \geq 10+8$
$\Rightarrow \quad O P+O Q \geq 18$
