Let `A-=(3,-4),B-=(1,2)dot` Let `P-=(2k-1,2k+1)` be a variable point such that `P A+P B` is the minimum. Then `k` is 7/9 (b) 0 (c) 7/8 (d) none of these

Options

A. `7//9`
B. 0
C. `7//8`
D. none of these

Answer

C

Explanation

We know that `PA+PBgeAB` (by triangle inequality). So, `PA+PB` is the minimum if `PA+PB=AB` , i.e., A,P,B are collinear. Therefore,
`|{:(3,,-4,,1),(1,,2,,1),(2k-1,,2k+1,,1):}|=0`
or `3(2-2k-1)+4(1-2k+1)+1(2k+1)+1(2k+1-4k+2)=0`
or `3-6k+8-8k+3-2k=0`
or `14-16k=0`
`therefore k=(7)/(8)`