Question

Let two vertices of triangle ABC be (2, 4, 6) and (0, –2, –5), and its centroid be (2, 1, –1). If the image of third vertex in the plane x + 2y + 4z = 11 is (α, β, γ), then αβ + βγ + γα is equal to

• A. 70
• B. 72
• C. 74
• D. 76

Answer 1

The Correct Option is C

Let's solve the problem step-by-step:

We are given two vertices of triangle ABC as \( A(2, 4, 6) \) and \( B(0, -2, -5) \), and the centroid \( G(2, 1, -1) \). Let the third vertex be \( C(x, y, z) \).

The formula for the centroid of a triangle with vertices \( (x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3) \) is:

\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)

Using the given data:

• \(\frac{2 + 0 + x}{3} = 2\) ⟹ \(2 + 0 + x = 6\) ⟹ \(x = 4\)
• \(\frac{4 + (-2) + y}{3} = 1\) ⟹ \(4 - 2 + y = 3\) ⟹ \(y = 1\)
• \(\frac{6 + (-5) + z}{3} = -1\) ⟹ \(6 - 5 + z = -3\) ⟹ \(z = -4\)

Thus, the coordinates of vertex \( C \) are \((4, 1, -4)\).

Next, to find the image of point \( C \) in the plane \( x + 2y + 4z = 11 \), we use the formula for the image of a point \( (x_1, y_1, z_1) \) in the plane \( ax + by + cz + d = 0 \):

\left(x - \frac{2a}{a^2 + b^2 + c^2}(ax_1 + by_1 + cz_1 + d), y - \frac{2b}{a^2 + b^2 + c^2}(ax_1 + by_1 + cz_1 + d), z - \frac{2c}{a^2 + b^2 + c^2}(ax_1 + by_1 + cz_1 + d)\right)

For the plane \( x + 2y + 4z = 11 \), we have \( a = 1 \), \( b = 2 \), \( c = 4 \), \( d = -11 \), and the point \( C(4, 1, -4) \).

Calculate \( ax_1 + by_1 + cz_1 + d \):

1(4) + 2(1) + 4(-4) - 11 = 4 + 2 - 16 - 11 = -21

Calculate the image coordinates:

• \(\alpha = 4 - \frac{2(1)(-21)}{1^2 + 2^2 + 4^2} = 4 + \frac{42}{21} = 4 + 2 = 6\)
• \(\beta = 1 - \frac{2(2)(-21)}{1^2 + 2^2 + 4^2} = 1 + \frac{84}{21} = 1 + 4 = 5\)
• \(\gamma = -4 - \frac{2(4)(-21)}{1^2 + 2^2 + 4^2} = -4 + \frac{168}{21} = -4 + 8 = 4\)

Thus, the image coordinates are \( (6, 5, 4) \).

Finally, compute \( \alpha \beta + \beta \gamma + \gamma \alpha \):

6 \times 5 + 5 \times 4 + 4 \times 6 = 30 + 20 + 24 = 74

The value of \( \alpha \beta + \beta \gamma + \gamma \alpha \) is \( \boxed{74} \).