Q21 of 45 Page 1

Find the coordinates of the point, where the line intersects the plane x - y + z - 5 = 0. Also find the angle between the line and the plane.

OR


Find the vector equation of the plane which contains the line of intersection of the planes and and which is perpendicular to the plane

The equation of the given line is

…(i)


Let



⇒ x – 2 = 3λ, y + 1 = 4λ and z – 2 = 2λ


⇒ x = 3λ + 2, y = 4λ – 1 and z = 2λ + 2 …(ii)


Any point on the given line is (3λ + 2, 4λ – 1, 2λ + 2)


If this point lies on the given plane x – y + z – 5 = 0, then


3λ + 2 – (4λ – 1) + 2λ + 2 – 5 = 0


⇒ 3λ + 2 – 4λ + 1 + 2λ + 2 – 5 = 0


⇒ λ = 0


Putting λ = 0 in eq. (ii), we get


x = 2, y = -1 and z = 2


∴ The point of intersection of the given line and plane is (2, -1, 2)


Now, we have to find the angle between the given line and plane


Let θ be the angle between the given line and plane.


The given line is parallel to and the given plane is normal to the vector


We know that,









OR


The given equation of planes are:


…(i)


…(ii)


Any plane through the line of intersection of the two given plane is



…(iii)


If this plane is perpendicular to the plane


Then, 5(1+ 2λ) + 3(2 + λ) – 6(3 – λ) = 0


⇒ 5 + 10λ + 6 + 3λ – 18 + 6λ = 0


⇒ 19

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