Question

If the perpendicular distance from \( (1,2,4) \) to the plane \( 2x + 2y - z + k = 0 \) is 3, then \( k \) is:

• A. \( 4 \)
• B. \( 7 \)
• C. \( 9 \)
• D. \( 19 \)

Hint:

The perpendicular distance formula is widely used in coordinate geometry problems.

Answer 1

The Correct Option is B

Step 1: Understand the Concept The perpendicular distance \( d \) from a point \( (x_1, y_1, z_1) \) to a plane given by \( Ax + By + Cz + D = 0 \) is calculated using the formula: \[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \] where: - \( A, B, C \) are the coefficients of \( x, y, z \) in the plane equation. - \( D \) is the constant term in the equation. - \( (x_1, y_1, z_1) \) is the given point.
Step 2: Identify Given Values The given plane equation is: \[ 2x + 2y - z + k = 0 \] Comparing with the standard form \( Ax + By + Cz + D = 0 \), we get: \[ A = 2, \quad B = 2, \quad C = -1, \quad D = k \] The given point is \( (1,2,4) \) and the perpendicular distance is \( d = 3 \).
Step 3: Apply the Perpendicular Distance Formula Substituting the values in the formula: \[ 3 = \frac{|(2 \times 1) + (2 \times 2) + (-1 \times 4) + k|}{\sqrt{2^2 + 2^2 + (-1)^2}} \]
Step 4: Compute the Magnitude Calculate the denominator: \[ \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \] Calculate the numerator: \[ (2 \times 1) + (2 \times 2) + (-1 \times 4) + k = 2 + 4 - 4 + k = 2 + k \]
Step 5: Solve for \( k \) \[ 3 = \frac{|2 + k|}{3} \] Multiply both sides by 3: \[ |2 + k| = 9 \] Solving for \( k \): \[ 2 + k = \pm 9 \]
Step 6: Find Possible Values of \( k \) \[ k = 9 - 2 = 7 \quad \text{or} \quad k = -9 - 2 = -11 \] Since \( k = 7 \) is present in the given options, we select: \[ \boxed{7} \]

Answer 2

To find the value of \( k \) for the plane \( 2x + 2y - z + k = 0 \), given that the perpendicular distance from the point \( (1, 2, 4) \) to the plane is 3, we use the formula for the distance \( d \) from a point \((x_1, y_1, z_1)\) to a plane \( Ax + By + Cz + D = 0 \):

\[ d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

Plugging in the values:

\[ A = 2, \, B = 2, \, C = -1, \, D = k, \, (x_1, y_1, z_1) = (1, 2, 4) \]

The formula becomes:

\[ 3 = \frac{|2 \cdot 1 + 2 \cdot 2 - 1 \cdot 4 + k|}{\sqrt{2^2 + 2^2 + (-1)^2}} \]

Simplifying the denominator:

\[ \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

The expression becomes:

\[ 3 = \frac{|2 + 4 - 4 + k|}{3} \]

\[ 3 = \frac{|2 + k|}{3} \]

Multiplying both sides by 3:

\[ 9 = |2 + k| \]

This leads to two cases:

1. \( 2 + k = 9 \)

\( k = 9 - 2 \)

\( k = 7 \)

2. \( 2 + k = -9 \)

\( k = -9 - 2 \)

\( k = -11 \)

Since \( k = -11 \) is not among the options provided, the correct value for \( k \) is:

7