A plane meets the coordinate axes in A,B,C such that the centroid of triangle ABC is the point `(p,q,r)`. If the equation of the plane is `x/p+y/q+z/r=k` then `k=`

Options

A. 1
B. 2
C. 3
D. none of these

Explanation

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the equation of the plane
The equation of the plane is given as:

\[ \frac{x}{p} + \frac{y}{q} + \frac{z}{r} = k \]

This is in the intercept form, where \(p\), \(q\), and \(r\) are the x, y, and z-intercepts of the plane, respectively.

### Step 2: Convert the equation to intercept form
We can rewrite the equation by dividing both sides by \(k\):

\[ \frac{x}{pk} + \frac{y}{qk} + \frac{z}{rk} = 1 \]

From this, we can identify the intercepts:
- The x-intercept is \(pk\)
- The y-intercept is \(qk\)
- The z-intercept is \(rk\)

### Step 3: Identify the points where the plane meets the axes
The points where the plane intersects the axes are:
- Point A (on x-axis): \(A(pk, 0, 0)\)
- Point B (on y-axis): \(B(0, qk, 0)\)
- Point C (on z-axis): \(C(0, 0, rk)\)

### Step 4: Find the centroid of triangle ABC
The centroid \(G\) of triangle \(ABC\) can be calculated using the formula:

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

Substituting the coordinates of points A, B, and C:

\[ G\left(\frac{pk + 0 + 0}{3}, \frac{0 + qk + 0}{3}, \frac{0 + 0 + rk}{3}\right) = \left(\frac{pk}{3}, \frac{qk}{3}, \frac{rk}{3}\right) \]

### Step 5: Set the centroid equal to the given point
According to the problem, the centroid is given as the point \((p, q, r)\). Therefore, we set:

\[ \frac{pk}{3} = p, \quad \frac{qk}{3} = q, \quad \frac{rk}{3} = r \]

### Step 6: Solve for \(k\)
From the equations:

1. \(\frac{pk}{3} = p\) implies \(k = 3\)
2. \(\frac{qk}{3} = q\) implies \(k = 3\)
3. \(\frac{rk}{3} = r\) implies \(k = 3\)

All equations consistently give \(k = 3\).

### Conclusion
Thus, the value of \(k\) is:

\[ \boxed{3} \]