Prove that pt. A (-2, 3, 5) , B (1,2, 3) and C (7,0,-1) are co-lenier and also fine the ratio in
Which c divides ab .

To determine if points \( A(-2, 3, 5) \), \( B(1, 2, 3) \), and \( C(7, 0, -1) \) are collinear and to find the ratio in which point \( C \) divides the line segment \( AB \), we can follow these steps:

### Step 1: Check for Collinearity
Points \( A \), \( B \), and \( C \) are collinear if the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are parallel, which happens if one vector is a scalar multiple of the other.

1. **Find the vector \( \overrightarrow{AB} \):**
   \[
   \overrightarrow{AB} = B - A = (1 - (-2), 2 - 3, 3 - 5) = (3, -1, -2)
   \]

2. **Find the vector \( \overrightarrow{AC} \):**
   \[
   \overrightarrow{AC} = C - A = (7 - (-2), 0 - 3, -1 - 5) = (9, -3, -6)
   \]

3. **Check if \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are parallel:**
   If \( \overrightarrow{AC} = k \cdot \overrightarrow{AB} \) for some scalar \( k \), the vectors are parallel. Dividing component-wise:
   \[
   k = \frac{9}{3} = \frac{-3}{-1} = \frac{-6}{-2} = 3
   \]
   Since all components give the same scalar \( k = 3 \), the vectors \( \overrightarrow{AB} \) and \( \overrightarrow{AC} \) are parallel, indicating that points \( A \), \( B \), and \( C \) are collinear.

### Step 2: Find the Ratio in Which \( C \) Divides \( AB \)
Since \( C \) is on the line segment \( AB \), we can use the section formula to determine the ratio \( k : 1 \) in which \( C \) divides \( AB \).

If \( C \) divides \( AB \) in the ratio \( k : 1 \), then the coordinates of \( C \) can be given by:
\[
C = \left( \frac{k \cdot x_B + x_A}{k + 1}, \frac{k \cdot y_B + y_A}{k + 1}, \frac{k \cdot z_B + z_A}{k + 1} \right)
\]
Substitute \( C(7, 0, -1) \), \( A(-2, 3, 5) \), and \( B(1, 2, 3) \) into the formula and solve for \( k \).

1. **For the \( x \)-coordinate:**
   \[
   7 = \frac{k \cdot 1 + (-2)}{k + 1}
   \]
   Multiply by \( k + 1 \):
   \[
   7(k + 1) = k - 2
   \]
   \[
   7k + 7 = k - 2
   \]
   \[
   6k = -9 \Rightarrow k = 3
   \]
Thus, \( C \) divides \( AB \)

Prove that pt. A (-2, 3, 5), B (1,2, 3) and C (7,0,-1) are collinear and also find the ratio in which C divides AB.

Learn how to prove that points A(-2, 3, 5), B(1, 2, 3), and C(7, 0, -1) are collinear and determine the ratio in which point C divides line segment AB. This solution uses vector geometry, collinearity checks, and the section formula. Suitable for Grades 10-12.

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