Question

If the distances of $P(x,y)$ from $A(-1,5)$ and $B(5,1)$ are equal, then

• A. $2x=y$
• B. $3x=2y$
• C. $3x=y$
• D. $2x=3y$

Answer 1

Answer: (B)

$3x=2y$

Let the points be $P(x,y)$, $A(-1,5)$ and $B(5,1)$

Distance between two points is $\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

$PA=\sqrt{(-1-x{)}^2+(5-y{)}^2}=\sqrt{x^2+1+2x+25+y^2-10y}=\sqrt{x^2+y^2+2x-10y+26}$

Therefore, $P{A}^2=x^2+y^2+2x-10y+26$

Similarly,

$PB=\sqrt{(5-x{)}^2+(1-y{)}^2}=\sqrt{25+x^2-10x+1+y^2-2y}=\sqrt{x^2+y^2-10x-2y+26}$

Therefore, $P{B}^2=x^2+y^2-10x-2y+26$

Since it is given that the distances are equal therefore $P{A}^2=P{B}^2$ that is

$x^2+y^2+2x-10y+26=x^2+y^2-10x-2y+26$

$2x-10y=-10x-2y$

$2x+10x=-2y+10y$

$12x=8y$

$3x=2y$