P is a fixed point `(a,a,a)` on a line through the origin equally inclined to the axes, then any plane through `P_|_` to OP, makes intercepts on the axes, the sum of whose reciprocals is equal to

Options

A. `alpha`
B. `3/2a`
C. `3a/2`
D. `1/a`

Answer

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

### Step 1: Identify the point P and the line through the origin
The fixed point \( P \) is given as \( (a, a, a) \). The line through the origin that is equally inclined to the axes can be represented with direction ratios of \( 1, 1, 1 \). Therefore, the equation of the line can be expressed as:
\[ \frac{x}{1} = \frac{y}{1} = \frac{z}{1} \]

**Hint:** Remember that a line equally inclined to the axes means it has equal direction ratios.

### Step 2: Find the equation of the plane
The plane that passes through point \( P \) and is perpendicular to the line \( OP \) can be derived from the normal vector. The normal vector to the plane is given by the direction ratios of the line \( OP \), which are also \( 1, 1, 1 \). The equation of the plane can be written as:
\[ 1(x - a) + 1(y - a) + 1(z - a) = 0 \]
This simplifies to:
\[ x + y + z - 3a = 0 \]
or equivalently:
\[ x + y + z = 3a \]

**Hint:** Use the point-normal form of the plane equation to derive the plane's equation.

### Step 3: Find the intercepts on the axes
From the equation \( x + y + z = 3a \), we can find the intercepts on the axes:
- For the x-intercept, set \( y = 0 \) and \( z = 0 \):
\[ x = 3a \quad \Rightarrow \quad \text{Intercept on x-axis} = 3a \]
- For the y-intercept, set \( x = 0 \) and \( z = 0 \):
\[ y = 3a \quad \Rightarrow \quad \text{Intercept on y-axis} = 3a \]
- For the z-intercept, set \( x = 0 \) and \( y = 0 \):
\[ z = 3a \quad \Rightarrow \quad \text{Intercept on z-axis} = 3a \]

**Hint:** To find the intercepts, set the other two variables to zero in the plane equation.

### Step 4: Calculate the sum of the reciprocals of the intercepts
The intercepts on the axes are \( 3a, 3a, 3a \). The sum of the reciprocals of these intercepts is:
\[ \frac{1}{3a} + \frac{1}{3a} + \frac{1}{3a} = \frac{3}{3a} = \frac{1}{a} \]

**Hint:** When calculating the sum of reciprocals, ensure you add the fractions correctly.

### Final Result
The sum of the reciprocals of the intercepts is:
\[ \frac{1}{a} \]

Thus, the answer is \( \frac{1}{a} \).