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.

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

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

Then

According to the question

PA = PB

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 units12x1y2-y3 + x2y3-y1 + x3y1-y2 = 10a4 + 2 + 3-2 - b + 5b - 4 = 206a - 6 - 3b + 5b - 20 = 206a+2b-26 = 206a + 2b - 26 = 20   or   6a + 2b - 26 = -206a + 2b = 46   or   6a + 2b = 63a + b = 23     .....2      or    3a + b = 3      .....3SOLUTION OF 1 AND 2 :Multiply 2 by 3, we get9a + 3b = 69    .....4Adding 1 and 4, we get10a = 70  a = 7Put a = 7 in 1, we get7 - 3b = 1  -3b = -6  b = 2So, a,b = 7,2SOLUTION OF 1 AND 3 :Multiply 3 by 3, we get9a + 3b = 9   ....5Adding 1 and 5, we get10a = 10  a = 1Put a = 1 in 1, we get1 - 3b = 1b = 0So, a,b = 1, 0So, coordinates of P are either 7, 2 or 1, 0.

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