The distance between the two points A and $\mathrm{A}^{\prime}$ which lie on $\mathrm{y}=2$ such that both the line segments AB and $\mathrm{A}^{\prime} \mathrm{B}$ (where B is the point (2, 3)) subtend angle $\frac{\pi}{4}$ at the origin, is equal to :

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Straight Lines

The distance between the two points A and

A^{'}

which lie on

y = 2

such that both the line segments AB and

A^{'} B

(where B is the point (2, 3)) subtend angle

\frac{π}{4}

at the origin, is equal to :

Answer: (C)

Let

A = (x, 2)

lie on the line

y = 2

. The point

B = (2, 3)

.

The slope of OB is

m_{1} = \frac{3}{2}

.
The slope of OA is

m_{2} = \frac{2}{x}

.

The angle between OA and OB is

\frac{π}{4}

, so:

tan \frac{π}{4} = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}} = 1

\frac{\frac{3}{2} - \frac{2}{x}}{1 + \frac{3}{x}} = 1

Solving:

\frac{3 x - 4}{2 ( x + 3 )} = 1

This gives two cases:

The distance between

A = (10, 2)

and

A^{'} = (- \frac{2}{5}, 2)

is:

∣10 - (- \frac{2}{5}) ∣ = \frac{52}{5}

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Single correct option