`A(-2, 2, 3) and B(13, -3, 13)` and L is a line through A.
Q. Coordinate of the line point P which divides the join of A and B in the ratio 2:3 internally are

Options

A. `((33)/(5), -(2)/(5), 9)`
B. `(4, 0, 7)`
C. `((32)/(5), -(12)/(5), (17)/(5))`
D. `(20, 0, 35)`

Answer

To find the coordinates of the point P that divides the line segment joining points A and B in the ratio 2:3 internally, we can use the section formula.

The section formula states that if a point P divides the line segment joining points A(x1, y1, z1) and B(x2, y2, z2) in the ratio m:n, then the coordinates of point P are given by:

\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]

### Step 1: Identify the coordinates of points A and B
- Point A has coordinates A(-2, 2, 3)
- Point B has coordinates B(13, -3, 13)

### Step 2: Assign the values for the section formula
- Let \( m = 2 \) (the part of A)
- Let \( n = 3 \) (the part of B)

### Step 3: Calculate the x-coordinate of point P
Using the section formula for the x-coordinate:

\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m+n} = \frac{2 \cdot 13 + 3 \cdot (-2)}{2 + 3} \]

Calculating this:

\[ x = \frac{26 - 6}{5} = \frac{20}{5} = 4 \]

### Step 4: Calculate the y-coordinate of point P
Using the section formula for the y-coordinate:

\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} = \frac{2 \cdot (-3) + 3 \cdot 2}{2 + 3} \]

Calculating this:

\[ y = \frac{-6 + 6}{5} = \frac{0}{5} = 0 \]

### Step 5: Calculate the z-coordinate of point P
Using the section formula for the z-coordinate:

\[ z = \frac{m \cdot z_2 + n \cdot z_1}{m+n} = \frac{2 \cdot 13 + 3 \cdot 3}{2 + 3} \]

Calculating this:

\[ z = \frac{26 + 9}{5} = \frac{35}{5} = 7 \]

### Final Result
The coordinates of point P that divides the line segment joining A and B in the ratio 2:3 internally are:

\[ P(4, 0, 7) \]

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