Question

If the points \( (1, 2, 3) \) and \( (2, -1, 0) \) lie on the opposite sides of the plane \( 2x + 3y - 2z = k \), then \( k \) is:

• A. \( k<1 \)
• B. \( k>2 \)
• C. \( k<1 \) or \( k>2 \)
• D. \( 1<k<2 \)

Hint:

For points to lie on opposite sides of a plane, the values of the plane equation for both points must have opposite signs when subtracted by \( k \).

Answer 1

The Correct Option is D

Step 1: Understanding the condition.
The points \( (1, 2, 3) \) and \( (2, -1, 0) \) lie on opposite sides of the plane if the sign of \( 2x + 3y - 2z - k \) is different for both points.
Step 2: Calculating the value for the first point.
Substitute \( (1, 2, 3) \) into the plane equation \( 2x + 3y - 2z = k \): \[ 2(1) + 3(2) - 2(3) = 2 + 6 - 6 = 2. \] So, for the point \( (1, 2, 3) \), the value is \( 2 \).
Step 3: Calculating the value for the second point.
Substitute \( (2, -1, 0) \) into the plane equation \( 2x + 3y - 2z = k \): \[ 2(2) + 3(-1) - 2(0) = 4 - 3 = 1. \] So, for the point \( (2, -1, 0) \), the value is \( 1 \).
Step 4: Analyzing the opposite sides condition.
For the points to lie on opposite sides of the plane, the values of \( 2 \) and \( 1 \) must have different signs when subtracted by \( k \). This occurs when \( 1<k<2 \), making the correct answer (4).
Conclusion.
The correct answer is (4) \( 1<k<2 \), as this satisfies the condition for the points lying on opposite sides of the plane.