Understanding the Problem: Finding the Ratio of Division

The question asks us to find the ratio in which the line given by the equation \(x + y = 4\) divides the line segment connecting point P with coordinates \((-1, 1)\) and point Q with coordinates \((5, 7)\). Let the point where the line \(x + y = 4\) intersects the line segment PQ be R. We need to find the ratio \(PR : RQ\).

Method 1: Using the Section Formula

The section formula helps us find the coordinates of a point that divides a line segment in a given ratio. Conversely, we can use it to find the ratio if we know the coordinates of the dividing point and the endpoints of the segment.

Let the point R divide the line segment PQ in the ratio \(m : n\). The coordinates of P are \((x_1, y_1) = (-1, 1)\) and the coordinates of Q are \((x_2, y_2) = (5, 7)\). According to the section formula, the coordinates of R are:

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

Substituting the coordinates of P and Q, we get the coordinates of R as:

\(R = \left( \frac{n(-1) + m(5)}{m+n}, \frac{n(1) + m(7)}{m+n} \right) = \left( \frac{-n + 5m}{m+n}, \frac{n + 7m}{m+n} \right)\)

Since point R lies on the line \(x + y = 4\), its coordinates must satisfy the equation of the line. So, we substitute the x and y coordinates of R into the equation \(x + y = 4\):

\(\left( \frac{-n + 5m}{m+n} \right) + \left( \frac{n + 7m}{m+n} \right) = 4\)

Now, we solve this equation for the ratio \(m/n\):

\(\frac{(-n + 5m) + (n + 7m)}{m+n} = 4\)

\(\frac{-n + 5m + n + 7m}{m+n} = 4\)

\(\frac{12m}{m+n} = 4\)

Multiply both sides by \((m+n)\):

\(12m = 4(m+n)\)

\(12m = 4m + 4n\)

Subtract \(4m\) from both sides:

\(12m - 4m = 4n\)

\(8m = 4n\)

To find the ratio \(m : n\), we can write \(m/n\):

\(\frac{m}{n} = \frac{4}{8} = \frac{1}{2}\)

Thus, the ratio \(m : n\) is \(1 : 2\). Since R divides PQ in the ratio \(m:n\), we have \(PR : RQ = m : n = 1 : 2\).

Method 2: Using the Formula for Ratio of Division by a Line

A line with equation \(ax + by + c = 0\) divides the line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(\lambda : 1\), where \(\lambda = -\frac{ax_1 + by_1 + c}{ax_2 + by_2 + c}\). The ratio of division is \(\lambda : 1\), which is equivalent to \(\lambda : 1 = m : n\), so \(m/n = \lambda\).

The equation of the line is \(x + y - 4 = 0\). Here, \(a = 1\), \(b = 1\), and \(c = -4\).

The coordinates of point P are \((x_1, y_1) = (-1, 1)\).

The coordinates of point Q are \((x_2, y_2) = (5, 7)\).

Let's calculate the values of \(ax_1 + by_1 + c\) and \(ax_2 + by_2 + c\).

For point P(-1, 1):

\(ax_1 + by_1 + c = 1(-1) + 1(1) + (-4) = -1 + 1 - 4 = -4\)

For point Q(5, 7):

\(ax_2 + by_2 + c = 1(5) + 1(7) + (-4) = 5 + 7 - 4 = 12 - 4 = 8\)

Now, substitute these values into the formula for \(\lambda\):

\(\lambda = -\frac{ax_1 + by_1 + c}{ax_2 + by_2 + c} = -\frac{-4}{8} = \frac{4}{8} = \frac{1}{2}\)

The ratio \(\lambda : 1\) is \(\frac{1}{2} : 1\), which is equivalent to \(1 : 2\). Since \(\lambda\) is positive, the line divides the segment internally.

This confirms that the point R divides the line segment PQ in the ratio \(PR : RQ = 1 : 2\).

Conclusion

Both methods yield the same result. The line \(x + y = 4\) cuts the line joining P(-1, 1) and Q(5, 7) at R in the ratio \(1 : 2\).

Revision Table: Coordinate Geometry Formulas for Line Segments

Additional Information: Internal vs. External Division

When a point R divides a line segment PQ, the division can be either internal or external.

• Internal Division: The point R lies between the endpoints P and Q on the segment. The ratio \(PR : RQ\) is positive. In this problem, the ratio \(1:2\) is positive, indicating internal division. This makes sense because the line \(x+y=4\) passes between the points P(-1,1) and Q(5,7). We can check this by substituting the points into the line equation: For P, -1+1 = 0 < 4. For Q, 5+7 = 12 > 4. Since the values are on opposite sides of 4, the line must cross the segment between P and Q.
• External Division: The point R lies outside the line segment PQ, on the line extending PQ. The ratio \(PR : RQ\) is negative when calculated using the formula \(-\frac{ax_1 + by_1 + c}{ax_2 + by_2 + c}\). The order of points matters; R could be on the extension beyond P or beyond Q.

Understanding whether the division is internal or external helps in visualizing the position of the point of intersection relative to the given line segment.