The coordinate of two points A and B are (3,4) and (5,-2) respectively Find the coordinate of point P - Maths - Coordinate Geometry

Question

The coordinate of two points A and B are (3,4) and (5,-2)
respectively Find the coordinate of point P if PA=PB The area of
triangle APB=10

Mayank Bhardwaj · Accepted Answer

The coordinates of points A and B are (3, 4) and (5, –2).

Let the coordinates of the point P be (a, b)

Then

$PA = \sqrt{(a - 3)^{2} + (b - 4)^{2}} PB = \sqrt{(a - 5)^{2} + (b + 2)^{2}}$

According to the question

PA = PB

$\Rightarrow \sqrt{(a - 3)^{2} + (b - 4)^{2}} = \sqrt{(a - 5)^{2} + (b + 2)^{2}}$

On squaring both sides, we get

(a – 3)² + (b – 4)² = (a – 5)² + (b + 2)²

⇒ a² + 9 – 6a + b² + 16 – 8b = a² + 25 – 10a + b²+ 4 + 4b

⇒ 9 – 6a + 16 – 8b – (25 – 10a + 4 + 4b) = 0

⇒ 9 – 6a + 16 – 8b – 25 + 10a – 4 – 4b = 0

⇒ 4a – 12b – 4 = 0

⇒ a – 3b – 1 = 0

⇒ a – 3b = 1 ...(1)

$Now, area of ∆ PAB = 10 sq units \Rightarrow \frac{1}{2} |x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})| = 10 \Rightarrow |a (4 + 2) + 3 (- 2 - b) + 5 (b - 4)| = 20 \Rightarrow |6 a - 6 - 3 b + 5 b - 20| = 20 \Rightarrow |6 a + 2 b - 26| = 20 \Rightarrow 6 a + 2 b - 26 = 20 or 6 a + 2 b - 26 = - 20 \Rightarrow 6 a + 2 b = 46 or 6 a + 2 b = 6 \Rightarrow 3 a + b = 23 . . . . . (2) or 3 a + b = 3 . . . . . (3) SOLUTION OF (1) AND (2) : Multiply (2) by 3, we get 9 a + 3 b = 69 . . . . . (4) Adding (1) and (4), we get 10 a = 70 \Rightarrow a = 7 Put a = 7 in (1), we get 7 - 3 b = 1 \Rightarrow - 3 b = - 6 \Rightarrow b = 2 So, (a, b) = (7, 2) SOLUTION OF (1) AND (3) : Multiply (3) by 3, we get 9 a + 3 b = 9 . . . . (5) Adding (1) and (5), we get 10 a = 10 \Rightarrow a = 1 Put a = 1 in (1), we get 1 - 3 b = 1 \Rightarrow b = 0 So, (a, b) = (1, 0) So, coordinates of P are either (7, 2) or (1, 0) .$

The coordinate of two points A and B are (3,4) and (5,-2) respectively. Find the coordinate of point P if PA=PB. The area of triangle APB=10.