Question

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

• A. 5
• B. 5
• C. 3
• D. 3

Answer 1

The correct option is B 5

Let A = (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)

$\therefore \frac{b-1}{2}=\frac{a+4}{2}\Rightarrow a-b=-5...........\left(i\right)$
Since, AB is perpendicular to the line y = x.
$\therefore$ Product of slope of AB and slope of the line y = x is equal to – 1.
$\Rightarrow \frac{b+1}{a-4}\times 1=-1\Rightarrow b+1=-a+4$
A + b = 3 …….(ii)
On solving Eqs. (i) and (ii), we get a = − 1 , b = 4
$\therefore$ Coordinates of the point A at (−1, 4).
$\therefore AB=\sqrt{(-1-4{)}^2+(4+1{)}^2}=5\sqrt{2}$
Hence, (b) is correct answer.