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If P be a point equidistant from points A (3,4) and B (5,-2) and area of `DeltaPAB` is 10 square units, then find the co-ordinates of point P.

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Let co-ordinates of point P be (x,y).
Now, PA = PB
`implies" "sqrt((x-3)^(2)+(y-4)^(2))=sqrt((x-5)^(2)+(y+2)^(2))`
`implies" "x^(2)+9-6x+y^(2)+16-8y =x^(2)+25-10x+y^(2)+4+4y`
`implies" "4x-12y=4`
`implies" "x-3y=1" "..(1)`
and area of `Delta =10`
`implies" "1/2[x(4+2)+3(-2-y)+5(y-4)=+-10`
`implies" "6x-6-3y+5y-20=+-20`
`implies" " 6x+2y-26+-20`
`implies" " 6x+2y-26=20" "or" "6x+2y-26=-20`
`implies" " 3x+y=23" "...(2)or" "3x+y=3" "...(3)`
From eqs. (1) and (2) From eqs. (1) and (3)
`" "x=7,y=2" "x=1,y=0`
`therefore` Required co-ordinates = (7,2) or (1,0)
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