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

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*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