The orthocenter of triangle whose vertices are `A(a,0,0),B(0,b,0)` and `C(0,0,c)` is `(k/a,k/b,k/c)` then k is equal to

Options

A. `(1/a^(2)+1/b^(2)+1/c^(2))^(-1)`
B. `(1/a+1/b+1/c)^(-1)`
C. `(1/a^(2)+1/b^(2)+1/c^(2))`
D. `(1/a+1/b+1/c)`

Explanation

To find the value of \( k \) for the orthocenter of the triangle with vertices \( A(a, 0, 0) \), \( B(0, b, 0) \), and \( C(0, 0, c) \), we will follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- \( A(a, 0, 0) \)
- \( B(0, b, 0) \)
- \( C(0, 0, c) \)

### Step 2: Determine the vectors
We need to find the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \):
- \( \overrightarrow{AB} = B - A = (0 - a, b - 0, 0 - 0) = (-a, b, 0) \)
- \( \overrightarrow{AC} = C - A = (0 - a, 0 - 0, c - 0) = (-a, 0, c) \)

### Step 3: Find the normal vector to the plane containing the triangle
The normal vector \( \mathbf{n} \) can be found using the cross product of \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \):
\[ \mathbf{n} = \overrightarrow{AB} \times \overrightarrow{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -a & b & 0 \\ -a & 0 & c \end{vmatrix} \]

Calculating the determinant:
\[ \mathbf{n} = \mathbf{i}(bc - 0) - \mathbf{j}(-ac - 0) + \mathbf{k}(0 - (-ab)) = (bc, ac, ab) \]

### Step 4: Find the orthocenter
The orthocenter \( H \) of the triangle can be found using the formula:
\[ H = \left( \frac{a^2}{a^2 + b^2 + c^2}, \frac{b^2}{a^2 + b^2 + c^2}, \frac{c^2}{a^2 + b^2 + c^2} \right) \]

### Step 5: Set the orthocenter equal to the given coordinates
We are given that the orthocenter is \( \left( \frac{k}{a}, \frac{k}{b}, \frac{k}{c} \right) \). Thus, we can set:
\[ \frac{a^2}{a^2 + b^2 + c^2} = \frac{k}{a} \]
\[ \frac{b^2}{a^2 + b^2 + c^2} = \frac{k}{b} \]
\[ \frac{c^2}{a^2 + b^2 + c^2} = \frac{k}{c} \]

### Step 6: Solve for \( k \)
From the first equation:
\[ k = \frac{a^3}{a^2 + b^2 + c^2} \]
From the second equation:
\[ k = \frac{b^3}{a^2 + b^2 + c^2} \]
From the third equation:
\[ k = \frac{c^3}{a^2 + b^2 + c^2} \]

### Step 7: Equate the expressions for \( k \)
Since all expressions for \( k \) must be equal, we can equate any two:
\[ \frac{a^3}{a^2 + b^2 + c^2} = \frac{b^3}{a^2 + b^2 + c^2} \implies a^3 = b^3 \implies a = b \]
Similarly, we can find that \( b = c \) and \( a = c \).

### Final Result
Thus, we conclude that:
\[ k = \frac{a^3}{a^2 + b^2 + c^2} \]