First slide

Equation of line in Straight lines

Question

A straight line L with negative slope passes through the points (8, 2) and cuts the positive coordinate axes at points P and Q. As L varies then the absolute minimum value of OP + OQ is (O is origin)

Difficult

A

28
B

15
C

18
D

10

Solution

Let the equation of the line L be
$y - 2 = m (x - 8), m < 0$
$Coordinates of P and Q are P (8 - \frac{2}{m}, 0) and Q (0, 2 - 8 m) .$
$\begin{array}{l} \begin{aligned} s o, O P + O Q & = 8 - \frac{2}{m} + 2 - 8 m \\ = 10 + \frac{2}{- m} + 8 (- m) \\ \geq 10 + 2 \sqrt{\frac{2}{- m} \times 8 (- m)} \geq 18 \end{aligned} \\ absolute min . value of O P + O Q = 18 . \end{array}$

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