Question

If the coordinates of two points A and B are (3,4) and (5,-2) respectively. Then, the coordinates of any point P, if PA = PB and area of $\Delta$ PAB = 10, are

• A. (7, 5), (1, 0)
• B. (7, 2), (1, 0)
• C. (7, 2), (- 1, 0)
• D. (2, 7), (1, 0)

Answer 1

The correct option is B

(7, 2), (1, 0)

$LetthecoordinatesofPbe(x,y).Then,PA=PB\Rightarrow P{A}^2=P{B}^2\Rightarrow (x-3{)}^2+(y-4{)}^2=(x-5{)}^2+(y+2{)}^2\Rightarrow x-3y-1=0\cdots \cdots \cdots \left(i\right)Now,Areaof\Delta PAB=10\Rightarrow \frac{1}{2}\left|\begin{array}{ccc}x& y& 1\\ 3& 4& 1\\ 5& -2& 1\end{array}\right|=\pm 10\Rightarrow 6x+2y-26=\pm 20\Rightarrow 6x+2y-46=0or6x+2y-6=0\Rightarrow 3x+y-23=0or3x+y-23=0Solvingx-3y-1=0and3x+y-23=0Wegetx=7,y=2Solvingx-3y-1=0and3x+y=30Wegetx=1,y=0Thus,thecordinatesofPare(7,2)or(1,0).Hence,\left(b\right)isthecorrectanswer.$