1.

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»

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-<klux>3Y</klux>=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)}}}


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