STATEMENT-1: The point `P(-22,28)`lies on the line `3x+2y+10=0` such that for the points `A(4,2)and B(2,4),|PA-PB|` is maximum. STATEMENT-2: A point P on `ax+by+c=0` such that `|PA-PB|` maximum is determined by the point of intersection of the line AB and `Ax+by+c=0.`

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

C

Explanation

Let the points A nad B lie on the same side of the given line `ax+by+c=0.` Then from triangle inequality we know that
`|PA-PB|leAB`
`impliesmax|PA-PB|=AB`
`impliesP,A,B` are collinear
`implies` P is the poinjt of intersection of the lines AB and PL.
For the stateement-1,

We have
`L-=3x+2y+10`
`L(4,2)=12+4+10=26gt0`
`P(2,4)=6+8+10=24gt0`
`implies` points A, B lie on the same side of
`3x+2y+10=0" "...(i)`
Now equation of AB is
`y-2=(4-2)/(2-4)(x-4)`
`impliesx+y-6=0" "....(ii)`
Solving (i) and (ii) we get
`x=-22amd y=28`
Thus `P(-22,28)` is the required point
`implies ` Statement-1 is true
Obviously statement-2 is not always true when points A and B lie on hte opposite sides of the line. Hence option (3) is correct.