Question

If the angle between the line \( 2(x + 1) = y = z \) and the plane \( 2x - y + \sqrt{2} z + 4 = 0 \) is \( \frac{\pi}{6} \), then the value of \( \lambda \) is:

• A. \( \frac{135}{7} \)
• B. \( \frac{45}{11} \)
• C. \( \frac{45}{7} \)
• D. \( \frac{135}{11} \)

Hint:

To find the angle between a line and a plane, use the formula: \[ \cos \theta = \frac{|\vec{d} \cdot \vec{n}|}{|\vec{d}| |\vec{n}|} \] where \( \vec{d} \) is the direction vector of the line, and \( \vec{n} \) is the normal vector of the plane.

Answer 1

The Correct Option is C

We are given the line equation: \[ 2(x + 1) = y = z \] The direction ratios for the line are: \[ \vec{d} = (2, 1, 1) \] The equation of the plane is: \[ 2x - y + \sqrt{2}z + 4 = 0 \] The normal vector for the plane is: \[ \vec{n} = (2, -1, \sqrt{2}) \] The angle \( \theta \) between the line and the plane is given by: \[ \cos \theta = \frac{|\vec{d} \cdot \vec{n}|}{|\vec{d}| |\vec{n}|} \] Given that the angle between the line and the plane is \( \frac{\pi}{6} \), we know: \[ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \] Now, substituting the values and solving the equation, we get the value of \( \lambda \): \[ \lambda = \frac{45}{7} \] Thus, the correct answer is \( \boxed{\frac{45}{7}} \).