Question

Let P be the plane passing through the points (5, 3, 0), (13, 3, –2) and (1, 6, 2). For α ∈ N, if the distances of the points A (3, 4, α) and B (2, α, a) from the plane P are 2 and 3 respectively, then the positive value of a is

• A. 5
• B. 6
• C. 3
• D. 4

Answer 1

The Correct Option is D

Step 1: Compute the normal vector of the plane 

The determinant is computed to find the normal vector of the plane. Expanding along the first row:

\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 8 & 0 & -2 \\ 4 & -3 & -2 \end{vmatrix} = \hat{i}(-6) + 8\hat{j} - 24\hat{k} \]

Simplifying the determinant gives the normal vector:

\[ \text{Normal vector of the plane: } 3\hat{i} - 4\hat{j} + 12\hat{k} \]

Step 2: Write the equation of the plane

The equation of the plane is determined using the normal vector and a point on the plane:

\[ 3x - 4y + 12z = 3 \]

Step 3: Find \(\alpha\) (distance from \(A(3,4,\alpha)\))

The distance formula is applied to the point \(A(3,4,\alpha)\) and equated to 2:

\[ \frac{|3(3) - 4(4) + 12\alpha - 3|}{\sqrt{3^2 + (-4)^2 + 12^2}} = 2 \]

Simplify the numerator:

\[ |9 - 16 + 12\alpha - 3| = |12\alpha - 10| \]

Simplify the denominator:

\[ \sqrt{3^2 + (-4)^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13 \]

The equation becomes:

\[ \frac{|12\alpha - 10|}{13} = 2 \]

Multiply through by 13:

\[ |12\alpha - 10| = 26 \]

Solving gives two possible values for \(\alpha\):

\[ 12\alpha - 10 = 26 \quad \Rightarrow \quad \alpha = 3 \] \[ 12\alpha - 10 = -26 \quad \Rightarrow \quad \alpha = -8 \]

\(\alpha = -8\) is rejected based on the given conditions or physical constraints. Thus:

\[ \alpha = 3 \]

Step 4: Find \(a\) (distance from \(B(2,3,a)\))

Similarly, the distance formula is applied to the point \(B(2,3,a)\) and equated to 3:

\[ \frac{|3(2) - 4(3) + 12a - 3|}{13} = 3 \]

Simplify the numerator:

\[ |6 - 12 + 12a - 3| = |12a - 9| \]

The equation becomes:

\[ \frac{|12a - 9|}{13} = 3 \]

Multiply through by 13:

\[ |12a - 9| = 39 \]

Solving gives two possible values for \(a\):

\[ 12a - 9 = 39 \quad \Rightarrow \quad a = 4 \] \[ 12a - 9 = -39 \quad \Rightarrow \quad a = -2.5 \]

\(a = -2.5\) is rejected based on the given conditions or physical constraints. Thus:

\[ a = 4 \]

Final Answer

The values are:

• \(\alpha = 3\)
• \(a = 4\)