Question

Let $P(x,y)$ is a point where $x$ satisfies $x^2-|x|-6=0$ and $y$ satisfies $y^2+|y|-6=0$, if $A(1,1)$, then $\sum (PA{)}^2$ is

• A. $55$
• B. $60$
• C. $35$
• D. $30$

Answer 1

The correct option is B $60$

Given : $x^2-|x|-6=0\Rightarrow |x{|}^2-|x|-6=0$
$\Rightarrow |x|=3$ or $|x|=-2$(reject)
$\therefore x=\pm 3$
and,$y^2+|y|-6=0$
$\Rightarrow |y|=2$ or $|y|=-3$(reject)
$\Rightarrow y=\pm 2$
$\therefore (x,y)=(3,2)$ or $(3,-2)$ or $(-3,2)$ or $(-3,-2)$
$\Rightarrow$ Possible distnaces from $(1,1)\text{to}(x,y)$ are $\sqrt{5}$ units or $\sqrt{13}$units or $\sqrt{17}$ units or $5$ units

$\therefore \sum (PA{)}^2=5+13+17+25=60$