Answers

The Brainliest Answer!
2015-04-01T11:22:31+05:30

This Is a Certified Answer

×
Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
Let slope = - m ,  where m > 0.
let point A be (8, 2).

equation of L :  y = - m x + c.    It passes through A.
                       2 = - m * 8 + c  =>  c = 2 + 8 m
     OP = x intercept , ie., value of x when y = 0.
           0 = - m * OP + c  = - m * OP + 2 + 8 m
         => OP = 2 / m + 8
     OQ = y intercept , ie., value of y when x = 0
           OQ = c = 2 + 8 m

 OP + OQ = 8 m + 10 + 2 / m
 derivative of (OP + OQ) wrt m :  8 - 2 / m²
   derivative = 0 when:  m = +1/2  or  -1/2.  we take only the positive value.
 
minimum value of OP + OQ = 4 + 10 + 4 = 18
=========================================
simpler method:

 The equation of L in the intercept form:  x/OP + y/OQ = 1
 We are given that OP and OQ are positive.  As L passes through A (8, 2):
        8/OP + 2/OQ = 1
        8 OQ + 2 OP = OP * OQ
        OP = 8 OQ / (OQ - 2)

The sum of intercepts :  OP + OQ = 8 OQ / (OQ - 2) + OQ

Derivative of OP + OQ wrt  OQ:  [ (OQ -2) * 8 - 8 OQ * 1 ] /(OQ - 2)²  + 1
             = 1 - 16 /(OQ - 2)²
Derivative is 0  when  (OQ - 2)² = 16
           OQ - 2 = +4 or -4
           OQ = 6  or  -2.  we take only the positive value.
Minimum value of  OP + OQ =  8 * 6/(6-2) + 6
               = 12 + 6 = 18

3 5 3