From the point (1,-2,3) lines are drawn to meet the sphere `x^2+y^2+z^2=4` and they are divided internally in the ratio 2 : 3 . The locus of the point of division is

Options

A. `5x^2+5y^2+5z^2 - 6 x + 12y + 22=0`
B. `5(x^2+y^2+z^2)=22`
C. `5x^(2) +5y^2+5z^2-2xy-3yz -zx-6x +12y + 5y +22=0`
D. `5x^2+5y^2+5z^2-6x +12y +18z +22=0`

Answer

To find the locus of the point of division of the lines drawn from the point \( P(1, -2, 3) \) to the sphere defined by the equation \( x^2 + y^2 + z^2 = 4 \), which are divided internally in the ratio \( 2:3 \), we can follow these steps:

### Step 1: Define the Points
Let \( Q(x_1, y_1, z_1) \) be a point on the sphere. The coordinates of \( Q \) must satisfy the equation of the sphere:
\[ x_1^2 + y_1^2 + z_1^2 = 4 \]

### Step 2: Use the Section Formula
The coordinates of the point \( D(x_2, y_2, z_2) \) that divides the segment \( PQ \) in the ratio \( 2:3 \) can be found using the section formula. The coordinates are given by:
\[ x_2 = \frac{3x_1 + 2 \cdot 1}{2 + 3} = \frac{3x_1 + 2}{5} \]
\[ y_2 = \frac{3y_1 + 2 \cdot (-2)}{2 + 3} = \frac{3y_1 - 4}{5} \]
\[ z_2 = \frac{3z_1 + 2 \cdot 3}{2 + 3} = \frac{3z_1 + 6}{5} \]

### Step 3: Substitute \( x_1, y_1, z_1 \) in terms of \( x_2, y_2, z_2 \)
From the equations for \( x_2, y_2, z_2 \), we can express \( x_1, y_1, z_1 \):
\[ 3x_1 = 5x_2 - 2 \implies x_1 = \frac{5x_2 - 2}{3} \]
\[ 3y_1 = 5y_2 + 4 \implies y_1 = \frac{5y_2 + 4}{3} \]
\[ 3z_1 = 5z_2 - 6 \implies z_1 = \frac{5z_2 - 6}{3} \]

### Step 4: Substitute into the Sphere Equation
Now substitute these expressions into the sphere equation:
\[ \left(\frac{5x_2 - 2}{3}\right)^2 + \left(\frac{5y_2 + 4}{3}\right)^2 + \left(\frac{5z_2 - 6}{3}\right)^2 = 4 \]
Multiplying through by \( 9 \) (to eliminate the denominators):
\[ (5x_2 - 2)^2 + (5y_2 + 4)^2 + (5z_2 - 6)^2 = 36 \]

### Step 5: Expand and Simplify
Expanding each term:
\[ (5x_2 - 2)^2 = 25x_2^2 - 20x_2 + 4 \]
\[ (5y_2 + 4)^2 = 25y_2^2 + 40y_2 + 16 \]
\[ (5z_2 - 6)^2 = 25z_2^2 - 60z_2 + 36 \]
Combining these:
\[ 25x_2^2 + 25y_2^2 + 25z_2^2 - 20x_2 + 40y_2 - 60z_2 + 56 = 36 \]
Simplifying:
\[ 25x_2^2 + 25y_2^2 + 25z_2^2 - 20x_2 + 40y_2 - 60z_2 + 20 = 0 \]

### Step 6: Divide by 5
Dividing the entire equation by 5 gives:
\[ 5x_2^2 + 5y_2^2 + 5z_2^2 - 4x_2 + 8y_2 - 12z_2 + 4 = 0 \]

### Final Locus Equation
The locus of the point of division \( D(x_2, y_2, z_2) \) is:
\[ 5x^2 + 5y^2 + 5z^2 - 4x + 8y - 12z + 4 = 0 \]