Question

If a plane meets the coordinate axes at \( A \), \( B \), and \( C \) in such a way that the centroid of \( \triangle ABC \) is at the point \( (1, 2, 3) \), then the equation of the plane is:

• A. \( \frac{x}{1} + \frac{y}{2} + \frac{z}{3} = 1 \)
• B. \( \frac{x}{3} + \frac{y}{6} + \frac{z}{9} = 1 \)
• C. \( \frac{x}{1} + \frac{y}{2} + \frac{z}{3} = \frac{1}{3} \)
• D. None of these

Hint:

To find the equation of a plane passing through the coordinate axes, use the formula \( \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \), where \( a \), \( b \), and \( c \) are the intercepts on the \( x \), \( y \), and \( z \)-axes, respectively. The centroid of a triangle can be used to find these intercepts.

Answer 1

The Correct Option is B

We are given that the plane meets the coordinate axes at points \( A \), \( B \), and \( C \). Let the coordinates of these points be: - \( A(a, 0, 0) \) on the \( x \)-axis,
- \( B(0, b, 0) \) on the \( y \)-axis,
- \( C(0, 0, c) \) on the \( z \)-axis.
The equation of the plane passing through the points \( A \), \( B \), and \( C \) can be written as: \[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1. \] Now, we are given that the centroid \( G \) of \( \triangle ABC \) lies at the point \( (1, 2, 3) \). The centroid of a triangle with vertices \( (x_1, y_1, z_1) \), \( (x_2, y_2, z_2) \), and \( (x_3, y_3, z_3) \) is given by the average of the coordinates of the vertices:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right). \] Substituting the coordinates of \( A( a, 0, 0 ) \), \( B(0, b, 0) \), and \( C(0, 0, c) \), the centroid is: \[ G\left( \frac{a + 0 + 0}{3}, \frac{0 + b + 0}{3}, \frac{0 + 0 + c}{3} \right) = \left( \frac{a}{3}, \frac{b}{3}, \frac{c}{3} \right). \] We are told that the centroid is at \( (1, 2, 3) \). Thus, we have the system of equations: \[ \frac{a}{3} = 1, \quad \frac{b}{3} = 2, \quad \frac{c}{3} = 3. \] Solving these equations gives: \[ a = 3, \quad b = 6, \quad c = 9. \] Step 2: Equation of the plane. Substitute \( a = 3 \), \( b = 6 \), and \( c = 9 \) into the equation of the plane: \[ \frac{x}{3} + \frac{y}{6} + \frac{z}{9} = 1. \] Thus, the equation of the plane is: \[ \frac{x}{3} + \frac{y}{6} + \frac{z}{9} = 1, \] and the correct answer is (b).