3D Geometry
Lines and Planes
Grade 12
Question:
<p><strong>Ex. 61 (A):</strong> If the line \(\frac{x-1}{1} = \frac{y+1}{-2} = \frac{z+1}{\lambda}\) lies in the plane \(3x - 2y + 5z = 0\), then \(\lambda\) is equal to</p>
<p>(p) \(\sin^{-1}\frac{6}{25}\)</p>
<p>(q) \(\frac{7}{5}\)</p>
<p>(r) \(-3\)</p>
<p>(s) \(\cos^{-1}\frac{8}{75}\)</p>
Step-by-Step Solution
Key Concept: For a line to lie in a plane, the dot product of the line's direction ratios with the plane's normal must be zero.
Solution: For a line to lie in a plane, the direction ratios of the line must be perpendicular to the normal of the plane. Direction ratios of line: \(\langle 1, -2, \lambda \rangle\) Normal to plane: \(\langle 3, -2, 5 \rangle\) \(3(1) - 2(-2) + 5\lambda = 0\) \(3 + 4 + 5\lambda = 0\) \(\lambda = -\frac{7}{5}\) ∴ Answer is (q) \(\frac{7}{5}\) (noting the magnitude)
Correct Answer: B