3 point O`(0,0),P(a,a^2),Q(-b,b^2)(agt0,bgt0)` are on the parabola `y=x^2`. Let `S_(1)` be the area bounded by the line PQ and parabola let `S_2` be the area of the `Delta OPQ`, the minimum value of `S_(1)//S_(2)` is

Options

A. `2//3`
B. `5//3`
C. 2
D. `73`

Answer

To solve the problem, we need to find the minimum value of the ratio \( \frac{S_1}{S_2} \), where \( S_1 \) is the area bounded by the line \( PQ \) and the parabola \( y = x^2 \), and \( S_2 \) is the area of triangle \( OPQ \).

### Step 1: Find the equation of line \( PQ \)
The points \( P(a, a^2) \) and \( Q(-b, b^2) \) lie on the parabola \( y = x^2 \). The slope of line \( PQ \) can be calculated as follows:

\[ \text{slope} = \frac{b^2 - a^2}{-b - a} = \frac{(b-a)(b+a)}{-(b+a)} = -(b-a) \]

Using the point-slope form of the line equation, we have:

\[ y - a^2 = -(b-a)(x - a) \]

Rearranging gives us:

\[ y = -(b-a)x + (b-a)a + a^2 \]

This simplifies to:

\[ y = -(b-a)x + ab \]

### Step 2: Set up the integral for \( S_1 \)
The area \( S_1 \) is given by the integral of the difference between the line and the parabola from \( x = -b \) to \( x = a \):

\[ S_1 = \int_{-b}^{a} \left( -(b-a)x + ab - x^2 \right) \, dx \]

### Step 3: Calculate the integral
Calculating the integral:

\[ S_1 = \int_{-b}^{a} \left( -(b-a)x + ab - x^2 \right) \, dx \]

This can be split into three separate integrals:

\[ S_1 = \int_{-b}^{a} -(b-a)x \, dx + \int_{-b}^{a} ab \, dx - \int_{-b}^{a} x^2 \, dx \]

Calculating each integral:

1. **First integral**:
\[ \int_{-b}^{a} -(b-a)x \, dx = -\frac{(b-a)}{2} \left[ x^2 \right]_{-b}^{a} = -\frac{(b-a)}{2} \left( a^2 - b^2 \right) = -\frac{(b-a)(a+b)}{2} \]

2. **Second integral**:
\[ \int_{-b}^{a} ab \, dx = ab \left[ x \right]_{-b}^{a} = ab(a + b) \]

3. **Third integral**:
\[ \int_{-b}^{a} x^2 \, dx = \frac{1}{3} \left[ x^3 \right]_{-b}^{a} = \frac{1}{3} (a^3 + b^3) \]

Combining these results gives us:

\[ S_1 = ab(a+b) - \frac{(b-a)(a+b)}{2} - \frac{1}{3}(a^3 + b^3) \]

### Step 4: Calculate \( S_2 \)
The area \( S_2 \) of triangle \( OPQ \) can be calculated using the determinant formula:

\[ S_2 = \frac{1}{2} \left| 0(a^2 - b^2) + a(0 - b^2) + (-b)(a^2 - 0) \right| = \frac{1}{2} \left| -ab^2 + ba^2 \right| = \frac{1}{2} ab(a - b) \]

### Step 5: Find the ratio \( \frac{S_1}{S_2} \)
Now we can find the ratio \( \frac{S_1}{S_2} \):

\[ \frac{S_1}{S_2} = \frac{ab(a+b) - \frac{(b-a)(a+b)}{2} - \frac{1}{3}(a^3 + b^3)}{\frac{1}{2} ab(a-b)} \]

### Step 6: Minimize the ratio
To find the minimum value of \( \frac{S_1}{S_2} \), we can use calculus or algebraic manipulation. The minimum value occurs when \( a = b \), leading to:

\[ \frac{S_1}{S_2} = \frac{2}{3} \]

### Final Answer
Thus, the minimum value of \( \frac{S_1}{S_2} \) is:

\[ \boxed{\frac{2}{3}} \]