The ratio in which the sphere `x^(2)+y^(2)+z^(2)=504` divides the line joining the points `(12, -4,8)` and `(27,-9,18)`is

Options

A. `-2:3`
B. 2:3
C. 3:4
D. 1:2

Answer

To find the ratio in which the sphere \(x^2 + y^2 + z^2 = 504\) divides the line joining the points \(A(12, -4, 8)\) and \(B(27, -9, 18)\), we can follow these steps:

### Step 1: Identify the coordinates of points A and B
The coordinates of point A are \(A(12, -4, 8)\) and the coordinates of point B are \(B(27, -9, 18)\).

### Step 2: Assume the ratio in which the line is divided
Let the ratio in which the line segment AB is divided by the point C be \(k:1\), where \(k\) is the ratio of segments AC to CB.

### Step 3: Use the section formula to find the coordinates of point C
The coordinates of point C, which divides the line segment AB in the ratio \(k:1\), can be calculated using the section formula:
\[ C\left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}, \frac{m z_2 + n z_1}{m+n}\right) \]
where \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\).

Substituting the coordinates of A and B:
\[ C\left(\frac{k \cdot 27 + 1 \cdot 12}{k + 1}, \frac{k \cdot (-9) + 1 \cdot (-4)}{k + 1}, \frac{k \cdot 18 + 1 \cdot 8}{k + 1}\right) \]
This gives us:
\[ C\left(\frac{27k + 12}{k + 1}, \frac{-9k - 4}{k + 1}, \frac{18k + 8}{k + 1}\right) \]

### Step 4: Substitute the coordinates of C into the sphere's equation
We know that point C lies on the sphere defined by the equation \(x^2 + y^2 + z^2 = 504\). Thus, we substitute the coordinates of C into this equation:
\[ \left(\frac{27k + 12}{k + 1}\right)^2 + \left(\frac{-9k - 4}{k + 1}\right)^2 + \left(\frac{18k + 8}{k + 1}\right)^2 = 504 \]

### Step 5: Simplify the equation
We will simplify each term:
1. \(\left(\frac{27k + 12}{k + 1}\right)^2\)
2. \(\left(\frac{-9k - 4}{k + 1}\right)^2\)
3. \(\left(\frac{18k + 8}{k + 1}\right)^2\)

Combine these terms and set the equation equal to 504. This will yield a quadratic equation in terms of \(k\).

### Step 6: Solve for k
After simplifying, we will solve the quadratic equation for \(k\).

### Step 7: Find the ratio
Once we have the value of \(k\), we can express the ratio \(m:n\) as \(k:1\).

### Final Step: Conclusion
After solving for \(k\), we find that the ratio in which the sphere divides the line segment is \(2:3\).