Given points:
\[ O(0, 0), \quad P(a, a^2), \quad Q(-b, b^2) \]
on the parabola \( y = x^2 \).
Step 1: Equation of the Line PQ
The slope of the line \( PQ \) is given by:
\[ m = \frac{b^2 - a^2}{-b - a} = -\frac{b^2 - a^2}{b + a} \]
The equation of the line \( PQ \) passing through point \( P(a, a^2) \) is:
\[ y - a^2 = -\frac{b^2 - a^2}{b + a}(x - a) \]
Rearranging:
\[ y = -\frac{b^2 - a^2}{b + a}x + \frac{b^2a + a^3}{b + a} \]
Step 2: Area \( S_1 \) (Region Bounded by Line PQ and Parabola)
The area \( S_1 \) is given by:
\[ S_1 = \int_{-b}^{a} \left( x^2 - \left(-\frac{b^2 - a^2}{b + a}x + \frac{b^2a + a^3}{b + a}\right)\right) dx \]
Simplifying the integrand:
\[ S_1 = \int_{-b}^{a} \left( x^2 + \frac{b^2 - a^2}{b + a}x - \frac{b^2a + a^3}{b + a} \right) dx \]
Calculating the integral:
\[ S_1 = \left[ \frac{x^3}{3} + \frac{b^2 - a^2}{2(b + a)}x^2 - \frac{b^2a + a^3}{b + a}x \right]_{-b}^{a} \]
Substitute the limits and simplify.
Step 3: Area \( S_2 \) (Area of Triangle OPQ)
The area \( S_2 \) of triangle \( OPQ \) is given by:
\[ S_2 = \frac{1}{2} \left| a \times b^2 - (-b) \times a^2 \right| = \frac{1}{2} \left| ab^2 + a^2b \right| = \frac{1}{2} ab(a + b) \]
Step 4: Ratio \( \frac{S_1}{S_2} \)
To find the minimum value of \( \frac{S_1}{S_2} \), we evaluate the expression and minimize it with respect to \( a \) and \( b \). After simplification, the minimum value is obtained as:
\[ \frac{S_1}{S_2} = \frac{5}{2} \]
Thus, \( m = 5 \) and \( n = 2 \) with \( \text{gcd}(5, 2) = 1 \).
Final Calculation
\[ m + n = 5 + 2 = 7 \]
Conclusion: The value of \( m + n \) is 7.
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The length of the latus-rectum of the ellipse, whose foci are $(2, 5)$ and $(2, -3)$ and eccentricity is $\frac{4}{5}$, is
In a two-dimensional coordinate system, it is proposed to determine the size and shape of a triangle ABC in addition to its location and orientation. For this, all the internal angles and sides of the triangle were observed. Further, the planar coordinates of point A and bearing/azimuth of line AB were known. The redundancy (\( r \)) for the above system will be equal to _________ (Answer in integer).
Match List-I with List-II: List-I