Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let A ≡ (a, b)

The coordinates of the midpoint 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 …… (1)

Since AB is $\perp$ to the line y = x,

$\therefore$ product of slope of AB and slope of the line y = x is equal to –1

$\Rightarrow \frac{b+1}{a-4}\times 1=-1\Rightarrow b+1=-a+4$

$\Rightarrow$ a + b = 3 …… (2)

Solving (1) and (2), we get a = –1, b = 4.

$\therefore$ Coordinates of the point A are (–1, 4)

$\therefore AB=\sqrt{(-1-4{)}^2+(4+1{)}^2}=5\sqrt{2}$