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If the distances of $P(x,\ y)$ from $A(5,\ 1)$ and $B(-1,\ 5)$ are equal, then prove that $3x = 2y$.
Given: The distances of $P(x,\ y)$ from $A(5,\ 1)$ and $B(-1,\ 5)$ are equal.
To do: To prove that $3x = 2y$.
Solution:
Given that $PA=PB$
$P( x,\ y) ,\ A( 5,\ 1)$ and $B( -1,\ 5)$
we know if there two points $(x_{1} ,\ y_{1})$ and $(x_{2} ,\ y_{2})$ ,
distance between the two points,$=\sqrt{( x_{2} -x_{1})^{2} +( y_{2} -y_{1})^{2}}$
using the above distance formula, we have
$PA=\sqrt{( x-5)^{2} +( y-1)^{2}}$
And $PB=\sqrt{( x+1)^{2} +( y-5)^{2}}$
As given $PA=PB$
$\sqrt{( x-5)^{2} +( y-1)^{2}}=\sqrt{( x+1)^{2} +( y-5)^{2}}$
$\Rightarrow ( x-5)^{2} +( y-1)^{2} =( x+1)^{2} +( y-5)^{2}$
$\Rightarrow x^{2} +25-10x+y^{2} +1-2y=x^{2} +1+2x+y^{2} +25-10y$
$\Rightarrow 26-10x-2y=2x-10y+26$
$\Rightarrow -10x-2x=-10y+2y$
$\Rightarrow -12x=-8y$
$\Rightarrow 3x=2y$
Updated on: 10-Oct-2022
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