Ifthe points `(1,1, lambda) and (-3,0,1)` are equidistant from the plane,`3x + 4y-12z + 13 = 0`, then `lambda` satisfiesthe equation

Options

A. `3x ^(2) -10x+ 7=0`
B. `3x ^(2) +10x + 7=0`
C. `3x ^(2) + 10x -13=0`
D. `3x^(2) -10x + 21=0`

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the points \( (1, 1, \lambda) \) and \( (-3, 0, 1) \) are equidistant from the plane given by the equation \( 3x + 4y - 12z + 13 = 0 \).

### Step-by-Step Solution

1. **Identify the points and the plane**:
- Let \( P(1, 1, \lambda) \) and \( B(-3, 0, 1) \) be the points.
- The equation of the plane is \( 3x + 4y - 12z + 13 = 0 \).

2. **Calculate the distance from point \( P \) to the plane**:
The distance \( d \) from a point \( (x_0, y_0, z_0) \) to the plane \( Ax + By + Cz + D = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
For point \( P(1, 1, \lambda) \):
- \( A = 3, B = 4, C = -12, D = 13 \)
- Plugging in the coordinates of \( P \):
\[ d_P = \frac{|3(1) + 4(1) - 12(\lambda) + 13|}{\sqrt{3^2 + 4^2 + (-12)^2}} \]
\[ = \frac{|3 + 4 - 12\lambda + 13|}{\sqrt{9 + 16 + 144}} = \frac{|20 - 12\lambda|}{\sqrt{169}} = \frac{|20 - 12\lambda|}{13} \]

3. **Calculate the distance from point \( B \) to the plane**:
For point \( B(-3, 0, 1) \):
\[ d_B = \frac{|3(-3) + 4(0) - 12(1) + 13|}{\sqrt{3^2 + 4^2 + (-12)^2}} \]
\[ = \frac{|-9 - 12 + 13|}{\sqrt{169}} = \frac{|-8|}{13} = \frac{8}{13} \]

4. **Set the distances equal**:
Since the points are equidistant from the plane, we have:
\[ \frac{|20 - 12\lambda|}{13} = \frac{8}{13} \]
Multiplying both sides by 13:
\[ |20 - 12\lambda| = 8 \]

5. **Solve the absolute value equation**:
This gives us two cases:
- Case 1: \( 20 - 12\lambda = 8 \)
- Case 2: \( 20 - 12\lambda = -8 \)

**For Case 1**:
\[ 20 - 12\lambda = 8 \implies 12\lambda = 20 - 8 \implies 12\lambda = 12 \implies \lambda = 1 \]

**For Case 2**:
\[ 20 - 12\lambda = -8 \implies 12\lambda = 20 + 8 \implies 12\lambda = 28 \implies \lambda = \frac{28}{12} = \frac{7}{3} \]

6. **Final values of \( \lambda \)**:
The values of \( \lambda \) that satisfy the equation are:
\[ \lambda = 1 \quad \text{and} \quad \lambda = \frac{7}{3} \]