Let the line `(x-2)/(3)=(y-1)/(-5)=(z+2)/(2)` lies in the plane `x+3y-alphaz+beta=0`. Then, `(alpha, beta)` equals

Options

A. `(6, -17)`
B. `(-6, 7)`
C. `(5, -15)`
D. `(-5, 15)`

Answer

To solve the problem, we need to find the values of \((\alpha, \beta)\) given that the line defined by

\[ \frac{x-2}{3} = \frac{y-1}{-5} = \frac{z+2}{2} \]

lies in the plane defined by

\[ x + 3y - \alpha z + \beta = 0. \]

### Step 1: Identify a point on the line
From the equation of the line, we can identify a point on the line. When \(t = 0\), we have:

\[ x = 2, \quad y = 1, \quad z = -2. \]

So, the point \(P(2, 1, -2)\) lies on the line.

**Hint:** To find a point on the line, set the parameter equal to zero.

### Step 2: Substitute the point into the plane equation
Since the line lies in the plane, the point \(P(2, 1, -2)\) must satisfy the plane equation. Substitute \(x = 2\), \(y = 1\), and \(z = -2\) into the plane equation:

\[ 2 + 3(1) - \alpha(-2) + \beta = 0. \]

This simplifies to:

\[ 2 + 3 + 2\alpha + \beta = 0. \]

Thus, we have:

\[ 2\alpha + \beta + 5 = 0. \]

Rearranging gives us our first equation:

\[ 2\alpha + \beta = -5. \quad \text{(Equation 1)} \]

**Hint:** Always substitute the coordinates of the point into the plane equation to check if it lies on the plane.

### Step 3: Find the direction ratios of the line and the plane
The direction ratios of the line are given by the coefficients in the equation:

\[ (3, -5, 2). \]

The direction ratios of the plane can be derived from the coefficients of \(x\), \(y\), and \(z\) in the plane equation:

\[ (1, 3, -\alpha). \]

**Hint:** Direction ratios can be identified from the coefficients of the variables in the equations.

### Step 4: Use the condition for coplanarity
For the line to lie in the plane, the scalar triple product of the direction ratios of the line and the plane must be zero:

\[ 3(1) + (-5)(3) + (-2)(-\alpha) = 0. \]

This simplifies to:

\[ 3 - 15 + 2\alpha = 0. \]

Thus, we have:

\[ 2\alpha - 12 = 0 \implies 2\alpha = 12 \implies \alpha = 6. \]

**Hint:** Use the condition of coplanarity to relate the direction ratios.

### Step 5: Substitute \(\alpha\) back into Equation 1
Now, substitute \(\alpha = 6\) back into Equation 1:

\[ 2(6) + \beta = -5. \]

This simplifies to:

\[ 12 + \beta = -5 \implies \beta = -5 - 12 = -17. \]

**Hint:** After finding one variable, substitute it back into the previous equations to find the other variable.

### Final Result
Thus, the values of \((\alpha, \beta)\) are:

\[ (\alpha, \beta) = (6, -17). \]

### Summary of Steps
1. Identify a point on the line.
2. Substitute the point into the plane equation.
3. Find the direction ratios of the line and the plane.
4. Use the condition for coplanarity.
5. Substitute back to find the other variable.