We are tasked with finding the ratio in which the line segment joining the points $A(3, -5)$ and $B(-1, 6)$ is divided by the line $y = x$.

Step-by-Step Solution:

1. Parametric form of the dividing point:

Let's assume that the line $y = x$ divides the line segment at a point $P(x_1, y_1)$ in the ratio $m : n$. The formula for the coordinates of a point dividing a line segment in a ratio $m : n$ is:

$P(x_1, y_1) = \left( \frac{m \cdot x_B + n \cdot x_A}{m + n}, \frac{m \cdot y_B + n \cdot y_A}{m + n} \right)$

For $A(3, -5)$ and $B(-1, 6)$, we have: $x_1 = \frac{m \cdot (-1) + n \cdot 3}{m + n}, \quad y_1 = \frac{m \cdot 6 + n \cdot (-5)}{m + n}$

Since the line $y = x$ passes through $P$, we know that $x_1 = y_1$. Thus, equating the two expressions for $x_1$ and $y_1$:

$\frac{m \cdot (-1) + n \cdot 3}{m + n} = \frac{m \cdot 6 + n \cdot (-5)}{m + n}$

2. Simplifying the equation:

Multiplying both sides by $m + n$ to eliminate the denominator:

$m \cdot (-1) + n \cdot 3 = m \cdot 6 + n \cdot (-5)$

Expanding both sides:

$- m + 3n = 6m - 5n$

Bringing all terms involving $m$ and $n$ to one side:

$- m - 6m = -5n - 3n$

$-7m = -8n$

This simplifies to:

$\frac{m}{n} = \frac{8}{7}$

3. Conclusion:

The line $y = x$ divides the line segment joining the points $(3, -5)$ and $(-1, 6)$ in the ratio 8:7.

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Would you like a detailed breakdown of any part of this solution or have any other questions?

Related Questions:

1. What is the formula to find the coordinates of the point dividing a line segment in a given ratio?
2. How do we derive the condition for a line passing through a specific point?
3. Can the ratio of division be negative, and what would that imply geometrically?
4. How does the line equation $y = x$ affect different line segments?
5. How would the solution change if a different line equation divided the segment?

Tip:

When solving ratio problems involving line segments and dividing lines, always start by identifying the general form of the dividing point and use known conditions to solve for unknowns.