Let the angles made with the positive -axis by two straight lines drawn from the point P(2, 3) and meeting the line at a distance from the point P be and . Then the value of is: [2026]

Question

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 $\sqrt{\frac{2}{3}}$ from the point P be $θ_{1}$ and $θ_{2}$ . Then the value of $(θ_{1} + θ_{2})$ is: [2026]

Smart Achievers · Accepted Answer

(1)

$Let Q is (\sqrt{\frac{2}{3}} \cos θ + 2, \sqrt{\frac{2}{3}} \sin θ + 3)$

$so, x + y = 6$

$\sqrt{\frac{2}{3}} (\cos θ + \sin θ) + 5 = 6$

$\sin θ + \cos θ = \sqrt{\frac{3}{2}}$

$1 + \sin 2 θ = \frac{3}{2}$

$\sin 2 θ = \frac{1}{2}$

$2 θ = \frac{π}{6}, \frac{5 π}{6}$

$θ = \frac{π}{12}, \frac{5 π}{6}$

$So θ_{1} + θ_{2} = \frac{π}{2}$

Solution

Q.
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 $\sqrt{\frac{2}{3}}$ from the point P be $θ_{1}$ and $θ_{2}$ . Then the value of $(θ_{1} + θ_{2})$ is: [2026]

1 $\frac{π}{2}$

2 $\frac{π}{3}$

3 $\frac{π}{6}$

4 $\frac{π}{12}$

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Solution

Q. 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 23 from the point P be θ1 and θ2. Then the value of (θ1+θ2) is: [2026]

1 π2

2 π3

3 π6

4 π12