Q. 275.0( 1 Vote )

# If lines and intersect, then find the value of k and hence find the equation of the plane containing these lines.[CBSE 2015]

Given the equation of line are- any random point on this line is given by (2λ+1, 3λ-1, 4λ+1)

Another line is: any random point on this line is given by (μ+3, 2μ+k, μ)

At point of intersection of the lines the random coordinates must coincide.

2λ + 1 = μ+3

2λ - μ = 2 …(1)

Also,

4λ + 1 = μ

μ = 4λ + 1 …(2)

Adding equation 1 and 2,we get-

2λ = 4λ + 3  As, 3λ – 1 = 2μ + k

we have the values of λ and μ.

k = 3λ – 2μ – 1

k = 3(-3/2) – 2(-5) – 1

k = 9/2

Now we need to find the equation of the plane containing these 2 lines.

For this we need the normal vector to plain and a point on plane.

For normal we need to take the cross product of direction ratios of line.

Direction ratio of line 1 is (2,3,4)

And direction ratio of line 2 is ( 1,2,1)

the normal vector is given as -   Let be any random vector on pthe lane. The equation of plane is given by – where a is any defined vector on planthe e.

As lines lie on a plane, so one of its points can be taken as a poi t on the plane.

(2λ+1, 3λ-1, 4λ+1) will give a point on putting λ = 0

Point is (1, -1, 1) Hence, the equation of the plane is: -5x + 2y + z = -5-2+1 = -6

Equation is: -5x+2y+z = -6

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos  Questions Based on 3D Geometry26 mins  Questions Based on 3D Geometry | Check Yourself40 mins  Revision of straight lines important formulas in one shot59 mins  Reminding 11th - Revision of Circles important formulas in 50 Minutes56 mins  Concept Builder Questions of straight Lines (Quiz Session)55 mins  Conic Section Part 164 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation view all courses RELATED QUESTIONS :

Find the coordinate of the point P where the line through and crosses the plane passing through three points and Also, find the ratio in which P divides the line segment AB.

Mathematics - Board Papers