In what ratio the point P(-2, y, z) divides the line joining the points A(2, 4, 3) and B(-4, 5, -6). Also, find the coordinates of point P.

Answer

To find the ratio in which the point P(-2, y, z) divides the line segment joining points A(2, 4, 3) and B(-4, 5, -6), and to find the coordinates of point P, we can use the section formula.

### Step 1: Identify the coordinates of points A and B
- A = (x1, y1, z1) = (2, 4, 3)
- B = (x2, y2, z2) = (-4, 5, -6)

### Step 2: Use the section formula for x-coordinate
The section formula states that if a point P divides the line segment joining A and B in the ratio m:n, then the coordinates of P can be given by:

\[ x = \frac{mx_2 + nx_1}{m+n} \]

Given that the x-coordinate of point P is -2, we can set up the equation:

\[ -2 = \frac{m(-4) + n(2)}{m+n} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:

\[ -2(m + n) = -4m + 2n \]

### Step 4: Simplify the equation
Expanding and rearranging gives:

\[ -2m - 2n = -4m + 2n \]

Bringing all terms involving m and n to one side results in:

\[ -2m + 4m = 2n + 2n \]

This simplifies to:

\[ 2m = 4n \]

### Step 5: Solve for the ratio m:n
Dividing both sides by 2 gives:

\[ m = 2n \]

Thus, the ratio m:n can be expressed as:

\[ \frac{m}{n} = \frac{2}{1} \]

### Step 6: Find the coordinates of point P
Now that we know the ratio is 2:1, we can find the y and z coordinates of point P using the section formula.

#### For y-coordinate:
Using the formula:

\[ y = \frac{my_2 + ny_1}{m+n} \]

Substituting the values:

\[ y = \frac{2(5) + 1(4)}{2 + 1} = \frac{10 + 4}{3} = \frac{14}{3} \]

#### For z-coordinate:
Using the formula:

\[ z = \frac{mz_2 + nz_1}{m+n} \]

Substituting the values:

\[ z = \frac{2(-6) + 1(3)}{2 + 1} = \frac{-12 + 3}{3} = \frac{-9}{3} = -3 \]

### Step 7: Compile the coordinates of point P
Thus, the coordinates of point P are:

\[ P = (-2, \frac{14}{3}, -3) \]

### Final Answer
The point P divides the line segment joining A and B in the ratio 2:1, and the coordinates of point P are:

\[ P(-2, \frac{14}{3}, -3) \]

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