Question

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

Answer 1

Let the coordinates of P be (x, y). Then,

PA = PB
Distance between the points is given by
$\sqrt{{(x_1-x_2)}^2+{(y_1-y_2)}^2}$

⇒ ${PA}^2$ = ${PB}^2$

⇒ ${(x-3)}^2$ + ${(y-4)}^2$ = ${(x-5)}^2$ + ${(y+2)}^2$

⇒ x - 3y - 1 = 0 -------(1)

Now, Area of ΔPAB = 10

⇒ $\frac{1}{2}$ |(4x + 3 × (-2) + 5y) - (3y + 20 - 2x)| = 10

⇒ |(4x + 5y - 6) - (2x + 3y + 20)| = 20

⇒ |6x + 2y - 26| = ± 20

⇒ 6x + 2y - 26 = ± 20

⇒ 6x + 2y - 46 = 0 or, 6x + 2y - 6 = 0

⇒ 3x + y - 23 = 0 or, 3x + y - 3 = 0

Solving x - 3y - 1 = 0 and 3x + y - 23 = 0 we get x = 7, y = 2.

Solving x - 3y - 1 = 0 and 3x + y - 3 = 0, we get x = 1, y = 0.

Thus, the coordinates of P are (7, 2) or (1, 0).