Three points $O(0, 0)$, $P(a, a^2)$, $Q(-b, b^2)$, $a>0, b>0$, are on the parabola $y = x^2$. Let $S_1$ be the area of the region bounded by the line $PQ$ and the parabola, and $S_2$ be the area of the triangle $OPQ$. If the minimum value of $\frac{S_1}{S_2}$ is $\frac{m}{n}$, $\gcd(m, n) = 1$, then $m + n$ is equal to:
The given matrix is:
\[
S_2 = \frac{1}{2} \begin{pmatrix} 0 & 0 & 1 \\ a & a^2 & 1 \\ -b & b^2 & 1 \end{pmatrix} = \frac{1}{2} (ab^2 + a^2b)
\]
Step 1: Equation for line \( PQ \):
The equation for \( PQ \) is given by:
\[
-y - a^2 = \frac{a^2 - b^2}{a + b} (x - a)
\]
Simplifying:
\[
y - a^2 = (a - b)x - (a - b)a
\]
\[
y = (a - b)x + ab
\]
Step 2: Integrating to find \( S_1 \):
Now we integrate:
\[
S_1 = \int_{-b}^{a} \left( (a - b)x + ab - x^2 \right) dx
\]
Breaking the integral:
\[
S_1 = (a - b) \frac{x^2}{2} + (ab)x - \frac{x^3}{3} \Bigg|_{-b}^{a}
\]
Simplifying:
\[
S_1 = (a - b) \left( \frac{a^2}{2} + (ab) - \frac{a^3}{3} \right) - \left( -\frac{b^3}{3} \right)
\]
\[
S_1 = (a - b)^2 \left( \frac{a + b}{2} \right) + ab(a + b) - \left( \frac{a^3 + b^3}{3} \right)
\]
From the relation \( m + n = 7 \), we conclude:
\[
m + n = 7
\]
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Approach Solution -2
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.
