Question:
Let (\alpha, \beta, \gamma) \text{ be the image of the point } P(3, 3, 5) \text{ in the plane } 2x + y - 3z = 6. \text{ Then } \alpha + \beta + \gamma \text{ is equal to:}
Let (\alpha, \beta, \gamma) \text{ be the image of the point } P(3, 3, 5) \text{ in the plane } 2x + y - 3z = 6. \text{ Then } \alpha + \beta + \gamma \text{ is equal to:}
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Remember, when finding the image of a point, use the formula for reflection across a plane. Pay close attention to the coefficients of the plane equation.
Updated On: Mar 22, 2025
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The Correct Option is C
Solution and Explanation
The equation of the plane is given as:
\[
2x + y - 3z = 6
\]
Let the point \( P(3, 3, 5) \) be the point whose image is \( (\alpha, \beta, \gamma) \). The image of a point in a plane can be found by using the formula for the reflection of a point across a plane.
The reflection formula is: \[ \alpha - x = \frac{2 \times \left(2x + y - 3z - 6 \right)}{2 + 1 + 1} \] Substitute the given values of \( x = 3, y = 3, z = 5 \), and calculate the values of \( \alpha, \beta, \gamma \). After solving for the reflection, we obtain: \[ \alpha = 6, \quad \beta = 5, \quad \gamma = -1 \] Now, calculate \( \alpha + \beta + \gamma \): \[ \alpha + \beta + \gamma = 6 + 5 - 1 = 10 \] Thus, the value of \( \alpha + \beta + \gamma \) is 10.
The reflection formula is: \[ \alpha - x = \frac{2 \times \left(2x + y - 3z - 6 \right)}{2 + 1 + 1} \] Substitute the given values of \( x = 3, y = 3, z = 5 \), and calculate the values of \( \alpha, \beta, \gamma \). After solving for the reflection, we obtain: \[ \alpha = 6, \quad \beta = 5, \quad \gamma = -1 \] Now, calculate \( \alpha + \beta + \gamma \): \[ \alpha + \beta + \gamma = 6 + 5 - 1 = 10 \] Thus, the value of \( \alpha + \beta + \gamma \) is 10.
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