Question

PA = PB and area of PAB = 10 sq. units. Find the coordinates of Pi coordinates of A and 3 are (1,2) and (3,8) respectively​

Answer 1

Answer :

The coordinates are (7,2).

Explanation :

Given that :

A and b are two points (3,4) , (5,-2) and PA=PB and area of triangle PAB= 10 square units.

To Find :

The coordinates of P.

Solution :

$\bigstar\;\textbf{\underline{\underline{Consider as - }}}$

Coordinate P as (x,y)

So,

It is given that PA = PB.

Now, Calculating distance PA. & PA = (x,y) (3,4).

$\sqrt{(3-x)^{2}+(4-y)^{2}}$

Now, Calculating distance PB. & PB = (x,y) (5,-2)

$\sqrt{(5-x)^{2}+(-2-y)^{2}}$

Now, We have :

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

Squaring both the sides in the equations,

$\implies \tt {(3-x)^{2}+(4-y)^{2}}={(5-x)^{2}+(-2-y)^{2}}\\ \\\implies9+x^{2}-6x+16+y^{2}-8y=25+x^{2}-10x+4+y^{2}+4y\\ \\\implies-6x-8y=-10x+4+4y\\ \\\implies 4x-12y=4\\ \\\implies x-3y=1..........(Equation 1)$

Area of triangle PAB = 10

Area of triangle of (3,4) (5,-2) and (x,y)

$\implies \tt Area\; of\; triangle\;PAB=\dfrac{1}{2}[3(-2-y)+5(y-4)+x(4+2)]=10\\ \\\implies -6+2y-20+6x=20\\ \\ \implies 46=2y+6x\\ \\ \implies 3x+y=23 ........(Equation 2)$

Now, solving equations 1 and 2,

So, x - 3y = 1  

∴ x = 3y + 1

From Equation 2,

$\implies \tt 3(3y+1)+y=23\\ \\\implies 9y+3+y=23\\ \\\implies 10y=20\\ \\\implies y= 2\\\ \\\implies x=3y+1\\ \\\implies x=(3 \times 2)+1\\ \\\implies x= 7\\ \\ \\ \\\bigstar\; \textbf{\underline{\underline{Therefore, the coordinates are (7,2)}}}$

Answer 2

Let coordinate P be x and y:-

PA = PB is according to the question

• So, (3,4) as (x and y)

$\sqrt{(3 - x) {}^{2} + (4 - y) {}^{2} }$

• So, (5, -2) as (x and y)

$\sqrt{(5 - x) {}^{2} + ( - x- y) {}^{2} }$

So ..... 3 , 4 and 5 , -2 both are equal.

Squaring both sides....

=(3 − x)² +(4 − y)²

= (5 − x)² +(−2 − y)²

= 9 + x² −6x + 16 + y 2 −8y²

= 25 + x² −10 x+ 4+ y 2 +4y²

= −6x − 8y = −10x+4+4y

= 4x − 12y = 4

= x−3y=1 Equation – (1)

10Area of triangle for PAB:-

(3,4) (5,-2) and (x,y)

21 is the area of triangle:-

= 3(−2−y)+5(y−4)+x(4+2)=10

= −6+2y−20+6x=20

= 46=2y+6x

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

Solving both equations:-

x - 3y = 1

x = 3y + 1

Equation 2

For (y)

= 3(3y+1)+y=23

= 9y+3+y=23

= 10y=20

= y=2

For (x)

=x=3y+1

= x=(3×2)+1

=x=7

Hence...... (7,2) is the coordinates.