Q. 204.0( 6 Votes )

Find the equation of the plane through the points (2,1,–1) and (–1,3,4), and perpendicular to the plane x – 2y + 4z = 10.

Answer :

It is given that,

A plane passes through the points (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x – 2y + 4z = 10.


We need to find the equation of such plane.


We know that,


The cartesian equation of a plane passing through (x1, y1, z1) having direction ratios proportional to a, b, c for its normal is


a(x – x1) + b(y – y1) + c(z – z1) = 0


So,


Let the equation of the plane passing through (2, 1, -1) be


a(x – 2) + b(y – 1) + c(z – (-1)) = 0


a(x – 2) + b(y – 1) + c(z + 1) = 0 …(i)


Since, it also passes through point (-1, 3, 4), just replace x, y, z by -1, 3, 4 respectively.


a(-1 – 2) + b(3 – 1) + c(4 + 1) = 0


-3a + 2b + 5c = 0 …(ii)


Since, a, b, c are direction ratios and this plane is perpendicular to the plane x – 2y + 4z = 10, just replace x, y, z by a, b, c (neglecting 10) respectively and equate to 0. So, we get


a – 2b + 4c = 0 …(iii)


If we need to solve two equations x1a + y1b + z1c = 0 and x2a + y2b + z2c = 0, the formula is:



Similarly, solve for equations (ii) and (iii).







That is,



a = 18λ



b = 17λ



c = 4λ


Substitute these values of a, b, c in equation (i),


a(x – 2) + b(y – 1) + c(z + 1) = 0


18λ(x – 2) + 17λ(y – 1) + 4λ(z + 1) = 0


λ[18(x – 2) + 17(y – 1) + 4(z + 1)] = 0


18(x – 2) + 17(y – 1) + 4(z + 1) = 0


18x – 36 + 17y – 17 + 4z + 4 = 0


18x + 17y + 4z – 36 – 17 + 4 = 0


18x + 17y + 4z – 49 = 0


18x + 17y + 4z = 49


Thus, equation of the required plane is 18x + 17y + 4z = 49.


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 GeometryQuestions Based on 3D GeometryQuestions Based on 3D Geometry26 mins
Questions Based on 3D Geometry | Check YourselfQuestions Based on 3D Geometry | Check YourselfQuestions Based on 3D Geometry | Check Yourself40 mins
Revision of straight lines important formulas in one shotRevision of straight lines important formulas in one shotRevision of straight lines important formulas in one shot59 mins
Reminding 11th - Revision of Circles important formulas in 50 MinutesReminding 11th - Revision of Circles important formulas in 50 MinutesReminding 11th - Revision of Circles important formulas in 50 Minutes56 mins
Concept Builder Questions of straight Lines (Quiz Session)Concept Builder Questions of straight Lines (Quiz Session)Concept Builder Questions of straight Lines (Quiz Session)55 mins
Conic Section Part 1Conic Section Part 1Conic 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
caricature
view all courses