the coordinates of a point P on the line 2x y + 5 = 0 such that |PA - Maths - Straight Lines

Question

the coordinates of a point P on the line 2x – y + 5 = 0 such
that |PA – PB | is maximum where A is ( 4, -2) and B is (2 ,
-4) will be what ?

Rishabh Mittal · Accepted Answer

Consider a point Px,y 
Let us consider that angle between line AP and BP is θ
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"https://s3mn%20mnimgs%20com/img/shared/ck-files/ck_5560d20ea3649%20jpg"
style="width: 476px; height: 318px;">
so, cos θ=AP2+BP2-AB22 AP
BPas cosθ≤1therefore,AP2+BP2-AB22 AP
BP≤1AP2+BP2-AB2≤2 AP BPAP2+BP2-2 AP
BP≤AB2since AB2=4-22+-2+42=4+4=8so,AP2+BP2-2 AP
BP≤8AP-BP≤22Maximum value of AP-BP is 22
Equation of line AB,y-2=4-22-4x-4y-2=2-2x-4-2y+4=2x-82x+2y-12=0x+y-6=0Point p lies on AB and given line 2x-y+5=0solving this two equations,x=y-52y-52+y-6=0y-5+2y2=63y-5=12y=173and x=173-52=17-1532=26=13   so , point P13,173  ans

the coordinates of a point P on the line 2x – y + 5 = 0 such that |PA – PB | is maximum where A is ( 4, -2) and B is (2 , -4) will be what ?