67. Which of the following point(s) lies(lie) on the plane 2x + 3y + z = 6?
(A) (0, 0, 6)
(B) (0, 2, 0)
(C) (1, 1, 1)
(D) (3, 0, 0)
How to Determine Whether a Point Lies on a Plane – Complete Concept and Detailed Solution
One of the simplest yet most important applications of the equation of a plane is checking whether a given point lies on that plane. Although the process is straightforward, students often make calculation mistakes or incorrectly substitute the coordinates. Therefore, understanding the mathematical principle behind the test is essential for solving such questions quickly and accurately.
The present question is a classic example where several points are given, and we must determine which of them satisfy the equation of the plane. Such questions are commonly used by examiners because they require conceptual understanding rather than lengthy calculations.
Correct Answer
Options (A), (B), and (D)
Understanding the Equation of a Plane
A plane in three-dimensional space is represented by an equation of the form
Ax + By + Cz + D = 0
where A, B, and C determine the orientation of the plane, while D determines its position in space.
The given equation
2x + 3y + z = 6
can also be written as
2x + 3y + z − 6 = 0.
A point lies on the plane if and only if its coordinates satisfy the equation exactly. Therefore, solving this question simply requires substituting the coordinates of each point into the plane equation.
Step 1: Test Option (A)
The point is
(0,0,6)
Substitute the coordinates into the equation:
2(0) + 3(0) + 6
= 0 + 0 + 6
= 6
The left-hand side equals the right-hand side.
Therefore, the point satisfies the equation.
Option (A) lies on the plane.
Step 2: Test Option (B)
The point is
(0,2,0)
Substitute into the equation:
2(0) + 3(2) + 0
= 0 + 6 + 0
= 6
Again, the equation is satisfied.
Option (B) also lies on the plane.
Step 3: Test Option (C)
The point is
(1,1,1)
Substitute into the equation:
2(1) + 3(1) + 1
= 2 + 3 + 1
= 6
The equation is satisfied exactly.
Option (C) also lies on the plane.
Step 4: Test Option (D)
The point is
(3,0,0)
Substitute into the equation:
2(3) + 3(0) + 0
= 6 + 0 + 0
= 6
The equation is satisfied.
Option (D) also lies on the plane.
Observation
After checking every option, we obtain the following results:
| Point | Value of 2x + 3y + z | Lies on Plane? |
|---|---|---|
| (0,0,6) | 6 | Yes ✔ |
| (0,2,0) | 6 | Yes ✔ |
| (1,1,1) | 6 | Yes ✔ |
| (3,0,0) | 6 | Yes ✔ |
Interestingly, every one of the four given points satisfies the equation of the plane.
Explanation of Every Option
Option (A): (0,0,6)
Substituting the coordinates gives 6, which matches the right-hand side of the plane equation. Therefore, this point lies on the plane.
Option (B): (0,2,0)
The value of 2x + 3y + z becomes 6, satisfying the equation exactly. Hence, this point lies on the plane.
Option (C): (1,1,1)
Substitution gives 2 + 3 + 1 = 6. Since the equality holds, this point also lies on the plane.
Option (D): (3,0,0)
Substituting the coordinates gives 6, confirming that this point also belongs to the plane.
Alternative Method
Instead of performing lengthy calculations, simply evaluate the expression 2x + 3y + z for each point. If the result equals 6, the point lies on the plane. Otherwise, it does not. Since each calculation here is very short, this method is extremely useful during competitive examinations where speed is important.
Geometrical Interpretation
A plane is an infinite flat surface extending in all directions in three-dimensional space. Every point whose coordinates satisfy the plane equation lies exactly on this surface. Any point that does not satisfy the equation lies either above or below the plane. Therefore, checking whether a point belongs to a plane is equivalent to verifying whether its coordinates satisfy the algebraic equation of the plane.
Related Practice Example
Determine whether the point (2,1,1) lies on the plane
x + 2y + 3z = 7.
Substituting the coordinates,
2 + 2(1) + 3(1)
= 2 + 2 + 3
= 7
Since the equation is satisfied, the point lies on the plane.
Practising similar examples improves both speed and accuracy in competitive examinations.
Key Takeaways
To determine whether a point lies on a plane, simply substitute its coordinates into the plane equation. If the left-hand side equals the right-hand side, the point lies on the plane. Otherwise, it does not. This simple substitution technique is one of the fastest methods for solving objective questions on planes.
Final Answer
Each of the given points satisfies the equation
2x + 3y + z = 6.
Therefore,
(0,0,6), (0,2,0), (1,1,1), and (3,0,0) all lie on the given plane.
Mathematically Correct Answer: (A), (B), (C), and (D)
