Let the angles made with the positive x-axis by two straight lines drawn from the point $\mathrm{P}(2,3)$ and meeting the line $\mathrm{x}+\mathrm{y}=6$ at a distance $\sqrt{\frac{2}{3}}$ from the point P be $\theta_{1}$ and $\theta_{2}$. Then the value of $\left(\theta_{1}+\theta_{2}\right)$ is :

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Straight Lines

Let the angles made with the positive x-axis by two straight lines drawn from the point

P (2, 3)

and meeting the line

x + y = 6

at a distance

\frac{2}{3}

from the point P be

θ_{1}

and

θ_{2}

. Then the value of

(θ_{1} + θ_{2})

is :

Answer: (C)

A point on the line through

P (2, 3)

inclined at angle

θ

to the positive x-axis, at distance

r = \frac{2}{3}

from

P

, is

(2 + r cos θ, 3 + r sin θ)

. For this point to lie on

x + y = 6

(2 + r cos θ) + (3 + r sin θ) = 6

, so

r (cos θ + sin θ) = 1

and

cos θ + sin θ = \frac{1}{r} = \frac{3}{2}

. Squaring,

1 + sin 2 θ = \frac{3}{2}

, hence

sin 2 θ = \frac{1}{2}

. The two admissible inclinations give

2 θ_{1} = \frac{π}{6}

and

2 θ_{2} = \frac{5 π}{6}

, so

θ_{1} = \frac{π}{12}

and

θ_{2} = \frac{5 π}{12}

. Therefore

θ_{1} + θ_{2} = \frac{π}{2}

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Single correct option