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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 $ΔPAB = 10$ is

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(7, 2)

(1, 0)

(- 7, 2)

(3, 0)

(7, - 2)

(5, 0)

(7, - 2)

(- 1, 0)

answer is A.

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Detailed Solution

Given that the coordinates of two points and B are

(3, 4)

and

(5, - 2)

respectively.
Let P

= (x, y)

such that

PA = PB .

\Rightarrow {PA}^{2} = {PB}^{2}

\Rightarrow {(x - 3)}^{2} + {(y - 4)}^{2} = {(x - 5)}^{2} + {(y + 2)}^{2}

\Rightarrow

x^{2} - 6 x + 9 + y^{2} - 8 y + 16 = x^{2} - 10 x + 25 + y^{2} + 4 y + 4

\Rightarrow

4 x - 12 y = 4

\Rightarrow

x - 3 y = 1

………………….(i)
Also, Area of

AB = 10

.
We know that Area of triangle whose vertices are (

x_{1}, y_{1}), (x_{2}, y_{2})

and (

x_{3}, y_{3})

\frac{1}{2} [x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]

sq. unit.
So we get the equation as follow,

| \frac{x (4 + 2) + 3 (- 2 - y) + 5 (y - 4)}{2} | = 10

\Rightarrow

| \frac{6 x - 6 - 2 y + 5 y - 20}{2} | = 10

\Rightarrow

\frac{6 x + 3 y - 26}{2} = \pm 10

\Rightarrow

6 x + 2 y - 26 = \pm 20

\Rightarrow

6 x + 2 y = 46

........(ii)
or,

6 x + 2 y = 6

........(iii)
To solve equation (i) and (iii) we must multiply (i) with 6 we get,

6 (x - 3 y) = 1 \times 6

6 x - 18 y = 6

…….(iv).
Now subtracting the equation (iii) from (iv) we get,

(6 x - 18 y)

6 x + 2 y) = 6 - 6

\Rightarrow 6 x - 18 y - 6 x - 2 y = 0

\Rightarrow - 20 y = 0

\Rightarrow y = 0

Now putting

y = 0

in the equation (iv) we get,

6 x - 6 \times 0 = 6

6 x = 6

x = \frac{6}{6}

x = 1

So, we get the coordinate is (1,0).
To solve equations (i) and (ii) we must multiply (i) with 6 we get,

6 x - 18 y = 6

……[from equation (iv)].
Now subtracting the equation (iv) from (ii) we get,

(6 x + 2 y) - (6 x - 18 y) = 46 - 6

\Rightarrow 20 y = 40

\Rightarrow y = \frac{40}{20}

\Rightarrow y = 2

Now putting the value y = 2 in the equation (i) we get,

x - 3 (2) = 1

\Rightarrow x - 6 = 1

\Rightarrow x = 6 + 1

\Rightarrow x = 7

So, we get the another coordinate is (7,2).
So, the coordinates of P are

(7, 2)

(1, 0)

.
So, the correct option is 1.

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