Question

If the point $A$ is symmetric to the point $B\equiv (4,-1)$ with respect to the bisector of the first quadrant, then the length of $AB$ is

• A. $5$
• B. $5\sqrt{2}$
• C. $3\sqrt{2}$
• D. $3$

Answer 1

The correct option is B $5\sqrt{2}$

Let $A\equiv (a,b).$

The coordinates of the mid point $C$ of $AB$ are $(\frac{a+4}{2},\frac{b-1}{2}).$

Since it lies on the line $y=x$ (the bisector of first quadrant)

$\therefore \frac{b-1}{2}=\frac{a+4}{2}\Rightarrow a-b=-5$

Since $AB$ is $\perp$ to the line $y=x,$ ...(1)
$\therefore$ product of slope of $AB$ and slope of the line $y=x$ is equal to $-1$ $\Rightarrow \frac{b+1}{a-4}\times 1=-1\Rightarrow b+1=-a+4$$\Rightarrow a+b=3$
Solving, we get $a=-1,b=4.$
$\therefore$ Coordinates of the point $A$ are $(-1,4),$
$\therefore AB=\sqrt{{(-1-4)}^2+{(4+1)}^2}=5\sqrt{2}$