According to the meaning of the question, k < 0

Then find the intersection. Under the combination of two functions.

2k/x=-x+√3k

-x？ +√3kx-2k=0

The sum of two roots is -b/a=√3, and the product of two roots is c/a=2k.

Then the OP distance is useless, draw the picture yourself,

You'll find it, op =x 1？ +y 1？

The intersection of inverse proportional function and linear function is characterized by x2 = y 1, x 1 = y2.

So op? =x 1？ +y 1？ =x 1？ +x2？ =(x 1+x2)？ -2x 1x2=(√3)？ -2×2k=7

Calculate k =-7

Digest well.