find the ratio in which the line segment joining the points (-3,10) and (6,-8) is divided by (-1,6).

Let the points \( A(-3, 10) \), \( B(6, -8) \), and \( P(-1, 6) \) be given. We need to find the ratio in which point \( P \) divides the line segment joining points \( A \) and \( B \).

Let the ratio be \( m:n \).

Using the section formula, the coordinates of the point dividing the line segment in the ratio \( m:n \) are given by:

\[
(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)
\]

Here, \( (x_1, y_1) = (-3, 10) \), \( (x_2, y_2) = (6, -8) \), and the coordinates of the point dividing the segment are \( P(-1, 6) \).

Thus, we have:
\[
-1 = \frac{m \cdot 6 + n \cdot (-3)}{m+n} \quad \text{(equation 1)}
\]
\[
6 = \frac{m \cdot (-8) + n \cdot 10}{m+n} \quad \text{(equation 2)}
\]

Now, solving these two equations for \( m \) and \( n \), let's proceed.The solution indicates that \( m = \frac{2n}{7} \). This means the ratio in which point \( P(-1, 6) \) divides the line segment joining points \( A(-3, 10) \) and \( B(6, -8) \) is \( 2:7 \).

Let me know if you'd like further explanation or have any questions!

Here are 5 related questions to deepen understanding:
1. How is the section formula derived?
2. What happens if the ratio is negative?
3. Can you apply the section formula for three-dimensional points?
4. How does the ratio change if point \( P \) is closer to point \( A \) than point \( B \)?
5. Can the section formula be used to find the midpoint of a line segment?

**Tip:** Always cross-check by plugging the ratio back into the section formula to verify correctness!

Find the ratio in which the line segment joining the points (-3,10) and (6,-8) is divided by (-1,6).

Learn how to find the ratio in which a point divides a line segment using coordinate geometry. This problem involves applying the section formula to determine the division ratio of the line segment between points (-3,10) and (6,-8) by the point (-1,6). Suitable for students in Grades 9-11.

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