For the parabola `y=-x^(2)`, let `a lt 0 and b gt 0,P(a, -a^(2)) and Q(b, -b^(2))`. Let M be the mid-point of PQ and R be the point of intersection of the vertical line through M, with the parabola. If the ratio of the area of the region bounded by the parabola and the line segment PQ to the area of the triangle PQR be `(lambda)/(mu)`, where `lambda and mu` are relatively prime positive integers, then find the value of `(lambda+mu)`:

Answer

To solve the problem step by step, we will follow the given information and derive the required areas and ratios.

### Step 1: Identify Points P and Q
Given the parabola \( y = -x^2 \):
- Point \( P(a, -a^2) \) where \( a < 0 \)
- Point \( Q(b, -b^2) \) where \( b > 0 \)

### Step 2: Find the Midpoint M of PQ
The coordinates of the midpoint \( M \) of points \( P \) and \( Q \) can be calculated as follows:
\[ M = \left( \frac{a + b}{2}, \frac{-a^2 - b^2}{2} \right) \]

### Step 3: Find the Vertical Line through M
The vertical line through \( M \) has the equation \( x = \frac{a + b}{2} \).

### Step 4: Find the Intersection Point R
To find point \( R \), we substitute \( x = \frac{a + b}{2} \) into the parabola equation:
\[ y = -\left(\frac{a + b}{2}\right)^2 = -\frac{(a + b)^2}{4} \]
Thus, the coordinates of point \( R \) are:
\[ R\left(\frac{a + b}{2}, -\frac{(a + b)^2}{4}\right) \]

### Step 5: Calculate Area of Triangle PQR
The area \( A_{PQR} \) of triangle \( PQR \) can be calculated using the formula:
\[ A_{PQR} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base \( PQ \) can be calculated as:
\[ PQ = \sqrt{(b - a)^2 + (-b^2 + a^2)^2} \]
The height is the vertical distance from \( R \) to line segment \( PQ \).

### Step 6: Calculate Area of Region Bounded by Parabola and Line Segment PQ
The area \( A_{bounded} \) between the parabola and the line segment \( PQ \) can be derived from the properties of parabolas:
\[ A_{bounded} = \frac{2}{3} \times A_{PQR} \]

### Step 7: Ratio of Areas
The ratio of the area of the region bounded by the parabola and the line segment \( PQ \) to the area of triangle \( PQR \) is:
\[ \frac{A_{bounded}}{A_{PQR}} = \frac{\frac{2}{3} A_{PQR}}{A_{PQR}} = \frac{2}{3} \]

### Step 8: Identify \( \lambda \) and \( \mu \)
Here, \( \lambda = 2 \) and \( \mu = 3 \). Since 2 and 3 are relatively prime positive integers, we can find:
\[ \lambda + \mu = 2 + 3 = 5 \]

### Final Answer
Thus, the value of \( \lambda + \mu \) is \( \boxed{5} \).