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 $\theta_1$ and $\theta_2$. Then the value of $(\theta_1+\theta_2)$ is

• A. $\dfrac{\pi}{6}$
• B. $\dfrac{\pi}{2}$
• C. $\dfrac{\pi}{12}$
• D. $\dfrac{\pi}{3}$

Hint:

When the product of slopes equals 1, the sum of the angles made with the $x$-axis is $\dfrac{\pi}{2}$.

Answer 1

The Correct Option is B

To find the value of \((\theta_1 + \theta_2)\) for 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 \(P\), follow these steps:

1. First, calculate the center point of the circle on the line \(x + y = 6\) that is at a distance of \(\sqrt{\frac{2}{3}}\) from \(P(2, 3)\). We know that the direction of the line perpendicular to \(x + y = 6\) is \((1, 1)\), so the line perpendicular from \(P\) to \(x + y = 6\) will be along this direction.
2. The perpendicular distance from a point \((x_1, y_1)\) to a line \(Ax + By + C = 0\) is given by:

\(\frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}\)

1. In our scenario, \(A = 1\), \(B = 1\), \(C = -6\), and \((x_1, y_1) = (2, 3)\). Calculate:

\(\frac{|1 \cdot 2 + 1 \cdot 3 - 6|}{\sqrt{1^2 + 1^2}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}}\)

1. Hence, we find the midpoints on the line which will be the points where the given distance forms a right angle with the line. Let these be \(P_1\) and \(P_2\).
2. Using the distance formula on each potential line from \(P(2, 3)\) to \(x + y = 6\):
3. Calculate for \(\theta_1\) and \(\theta_2\) using the possible directions of the perpendicular, understanding that \(1\) and \(m\) are roots satisfying:

\(m_1 \cdot m_2 = -1\)

1. This leads us to conclude \(\theta_1 + \theta_2 = \frac{\pi}{2}\) since the sum of angles in two perpendicular lines is always \(\frac{\pi}{2}\).

Conclusion: The sum of angles \((\theta_1 + \theta_2)\) that satisfy this condition is \(\frac{\pi}{2}\).