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 OP+OQ, as L varies, where O is the origin. (2002,5M)

Show Answer

Solution:

  1. Let L:(y−2)=m(x−8),m<0

The ponts P and Q are 8−2m,0 and (0,2−8m), respectively.

Then, OP+OQ=8−2m+(2−8m)=10+−2m+(−8m)

[using AM≥GM ]

⇒2−m+(−8m)≥216[∵2m and −8m are positive ]

⇒−2m+8m≥8

⇒10−2m+8m≥10+8

⇒OP+OQ≥18

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