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Question

If the co-ordinates of two points A and B are (3,4) and (5,-2) respectively, find the co-ordinates of any point P if PA=PB and Area of triangle PAB =10

A
(2,7)
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B
(0,1)
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C
(7,2)
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D
(1,0)
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Solution

The correct option is D (1,0)
Let the co-ordinates of P be (x,y). Then

PA=PB

PA2 = PB2

(x3)2+(y4)2=(x5)2+(y+2)2

x2+96x+y2+168y=x2+2510x+y2+4+4y

x - 3y - 1 = 0---------------(1)

now,

areaofPAB=12∣ ∣521471741∣ ∣ = 10

12|3(2y)+5(y4)+x(4+2)|

12|63y+5y20+6x|=10

12|6x+2y26|=±20

6x + 2y - 26 = 20 or 6x + 2y - 6 = 0

3x + y - 23 = 0 or 3x + y - 3 = 0

3x + y - 23 = 0 ------------------(2)

3x + y - 3 = 0 ------------------(2)

Solving equation 1 and 2

x - 3y - 1 = 0

3x + y - 23 = 0

Multiplying 3 in equation (2) and add it to equation (1)

10x = 70 x=7

Substituting x = 7 in equation 2, we get, y=2

Solving equation 1 and 3

x - 3y - 1 = 0

3x + y - 3 = 0

Multiply 3 in equation 3 and add it to equation 2,

10x = 10 x=1

Substituting x = 1, in equation 2

We get, y = 0

Thus, the co-ordinates of P is (7,2) and (1,0)

So, option C and D are correct.


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