Find the ratio in which y-axis divides the line segment joining the points A(5, -6) and 

 B (-1, -4). Also find the coordinates of the point of division.

To find the ratio in which the y-axis divides the line segment joining the points \( A(5, -6) \) and \( B(-1, -4) \), and to find the coordinates of the point of division, we can use the section formula.

The section formula states that the coordinates of a point \( P \) dividing the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) are given by:

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

Since the y-axis divides the segment, the x-coordinate of the point of division is \( 0 \). Let's denote the ratio in which the y-axis divides the segment by \( k:1 \). Then, using the section formula for the x-coordinate, we have:

\[
0 = \frac{k(-1) + 1(5)}{k + 1}
\]

Simplify and solve for \( k \):

\[
0 = \frac{-k + 5}{k + 1}
\]

Multiplying both sides by \( k + 1 \):

\[
0 = -k + 5
\]

Thus,

\[
k = 5
\]

So, the y-axis divides the line segment in the ratio \( 5:1 \).

Now, let's find the coordinates of the point of division. Using the section formula for the y-coordinate:

\[
y = \frac{5(-4) + 1(-6)}{5 + 1} = \frac{-20 - 6}{6} = \frac{-26}{6} = -\frac{13}{3}
\]

Therefore, the coordinates of the point of division are:

\[
\left( 0, -\frac{13}{3} \right)
\]

In summary:
- The ratio in which the y-axis divides the line segment joining the points \( A(5, -6) \) and \( B(-1, -4) \) is \( 5:1 \).
- The coordinates of the point of division are \( \left( 0, -\frac{13}{3} \right) \).

Would you like further details or have any questions?

Here are 8 related questions to explore further:
1. How does the section formula change if the ratio is given as a fraction instead of an integer?
2. Can you verify the ratio by considering the distances from the y-axis?
3. What happens if the dividing line is not the y-axis but a different vertical line?
4. How do you find the coordinates of the point dividing a segment in a given ratio on the x-axis?
5. What are the applications of the section formula in real-life scenarios?
6. How do you find the coordinates of a point dividing a segment in a given ratio in 3D space?
7. Can you derive the section formula from the basic principles of coordinate geometry?
8. How do you solve similar problems where the dividing point lies on a line other than the coordinate axes?

**Tip:** When working with ratios in coordinate geometry, always make sure to express the ratio clearly and verify your results by substituting back into the original formula or context.

Learn how to find the ratio in which the y-axis divides the line segment joining points A(5, -6) and B(-1, -4). Discover the coordinates of the division point using the section formula in coordinate geometry. Suitable for Grades 9-12.

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