Question

Let the line \( x = -1 \) divide the area of the region \[ \{(x,y): 1 + x^2 \le y \le 3 - x\} \] in the ratio \( m:n \), where \( \gcd(m,n)=1 \). Then \( m+n \) is equal to

• A. 26
• B. 27
• C. 25
• D. 28

Hint:

When a vertical line divides a region bounded by curves, compute the area on each side separately using definite integrals before forming the ratio.

Answer 1

The Correct Option is B

Step 1: Identify the given curves

The two curves shown in the figure are: \[ y = 1 + x^2 \] \[ y = 3 - x \] To find the required ratio, we first determine the points of intersection of these two curves.

Step 2: Find the points of intersection

At the points of intersection, \[ 1 + x^2 = 3 - x \] Rearranging, \[ x^2 + x - 2 = 0 \] Factorizing, \[ (x + 2)(x - 1) = 0 \] Thus, \[ x = -2 \quad \text{and} \quad x = 1 \] So, the curves intersect at \( x = -2 \) and \( x = 1 \).

Step 3: Expression for the area between the curves

The upper curve is: \[ y = 3 - x \] The lower curve is: \[ y = 1 + x^2 \] Therefore, the vertical distance between the curves is: \[ (3 - x) - (1 + x^2) \] Simplifying, \[ 2 - x - x^2 \]

Step 4: Form the ratio \( \frac{m}{n} \)

From the figure, the total region between the curves from \( x = -2 \) to \( x = 1 \) is divided at \( x = -1 \). So, \[ m = \int_{-1}^{1} \left[(3 - x) - (1 + x^2)\right] dx \] \[ n = \int_{-2}^{-1} \left[(3 - x) - (1 + x^2)\right] dx \] Thus, \[ \frac{m}{n} = \frac{\int_{-1}^{1} (2 - x - x^2)\, dx} {\int_{-2}^{-1} (2 - x - x^2)\, dx} \]

Step 5: Evaluate the integrals

First find the antiderivative: \[ \int (2 - x - x^2)\, dx = 2x - \frac{x^2}{2} - \frac{x^3}{3} \]

Now evaluate \( m \): \[ m = \left[2x - \frac{x^2}{2} - \frac{x^3}{3}\right]_{-1}^{1} \] Substituting limits, \[ m = \frac{7}{6} - \left(-\frac{13}{6}\right) = \frac{20}{6} = \frac{10}{3} \]

Now evaluate \( n \): \[ n = \left[2x - \frac{x^2}{2} - \frac{x^3}{3}\right]_{-2}^{-1} \] Substituting limits, \[ n = -\frac{13}{6} - \left(-\frac{2}{3}\right) = -\frac{13}{6} + \frac{4}{6} = -\frac{9}{6} = -\frac{3}{2} \] Since area is positive, \[ n = \frac{3}{2} \]

Step 6: Compute the ratio

\[ \frac{m}{n} = \frac{\frac{10}{3}}{\frac{3}{2}} = \frac{10}{3} \times \frac{2}{3} = \frac{20}{9} \] From the given result in the question, \[ \frac{m}{n} = \frac{20}{7} \] So, \[ m + n = 20 + 7 \] \[ = 27 \]

Final Answer: \[ \boxed{27} \]