Question

Suppose the line $\frac{x - 2}{\alpha} = \frac{y - 2}{-5} = \frac{z + 2}{2}$ lies on the plane $x + 3y - 2z + \beta = 0$. Then $(\alpha + \beta)$ is equal to _________.

Hint:

If a line lies on a plane, remember: "Direction is perpendicular to normal (\( \vec{d} \cdot \vec{n} = 0 \)) AND Point is on the plane". Always check both conditions to solve for two unknowns.

Answer 1

Correct Answer: 7

Step 1: Understanding the Concept:
For a straight line to lie entirely within a plane, two fundamental conditions must be satisfied:
1. The line must be parallel to the plane. This implies that the direction vector of the line must be perpendicular to the normal vector of the plane. \( (\text{Dot product } = 0) \).
2. Any arbitrary point on the line must also lie on the plane. Generally, we use the specific point given in the symmetric form of the line equation.
Step 2: Key Formula or Approach:
Given a line \( L: \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \) and a plane \( \Pi: ax + by + cz + d = 0 \).
If \( L \) lies on \( \Pi \), then:
1. \( al + bm + cn = 0 \)
2. \( ax_1 + by_1 + cz_1 + d = 0 \)
Step 3: Detailed Explanation:
The given line is:
\[ \frac{x - 2}{\alpha} = \frac{y - 2}{-5} = \frac{z + 2}{2} \] From the equation, the direction vector of the line is \( \vec{b} = \alpha \hat{i} - 5 \hat{j} + 2 \hat{k} \) and it passes through the point \( P(2, 2, -2) \).
The given plane is:
\[ x + 3y - 2z + \beta = 0 \] The normal vector to the plane is \( \vec{n} = \hat{i} + 3 \hat{j} - 2 \hat{k} \).

Applying condition 1 (perpendicularity):
\[ \vec{b} \cdot \vec{n} = 0 \] \[ (\alpha)(1) + (-5)(3) + (2)(-2) = 0 \] \[ \alpha - 15 - 4 = 0 \] \[ \alpha - 19 = 0 \] \[ \alpha = 19 \]
Applying condition 2 (point on the plane):
Substitute point \( P(2, 2, -2) \) into the plane equation:
\[ 2 + 3(2) - 2(-2) + \beta = 0 \] \[ 2 + 6 + 4 + \beta = 0 \] \[ 12 + \beta = 0 \] \[ \beta = -12 \]
We need to find the value of \( (\alpha + \beta) \):
\[ \alpha + \beta = 19 + (-12) \] \[ \alpha + \beta = 7 \] Step 4: Final Answer:
The value of \( (\alpha + \beta) \) is 7.