First slide

Cartesian plane

Question

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

Moderate

A

12
B

14
C

18
D

20

Solution

The equation of the line is
(y – 2) = m(x – 8), m < 0
The coordinates of P and Q are $P (8 - \frac{2}{m}, 0) a n d Q (0, 2 - 8 m)$
Therefore, $O P + O Q = 8 - \frac{2}{m} + 2 - 8 m$
$= 10 + \frac{2}{- m} + 8 (- m)$
$\geq 10 + 2 \sqrt{\frac{2}{- m} \times 8 (- m)} = 18$
Thus, absolute minimum value of OP + OQ = 18.

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