Step 1. The equation of the parabola is \(x^2 = 8y\)
Step 2. The chord with midpoint \( (x_1, y_1) \) has the equation \( T = S_1 \):
\(x x_1 - 4(y + y_1) = x_1^2 - 8y_1\)
Substituting \( (x_1, y_1) = (1, \frac{5}{4}) \):
\(x - 4 \left( y + \frac{5}{4} \right) = 1 - 8 \cdot \frac{5}{4} = -9\)
\(x - 4y = -4\)
Step 3. Since \( P(\alpha, \beta) \) lies on this chord and also on the parabola \( y = \frac{x^2}{4} \), we have:
\(\alpha - 4\beta = -4\)
\(\beta^2 = 4\alpha\)
Step 4. Solve equations (ii) and (iii):
Substitute \( \alpha = \frac{\beta^2}{4} \) from (iii) into (ii):
\(\frac{\beta^2}{4} - 4\beta = -4\)
\(\beta^2 - 16\beta + 16 = 0\)
\((\beta - 8)^2 = 48\)
\(\beta = 8 \pm 4\sqrt{3}\)
\(\beta = 8 \pm 4\sqrt{3}\)
Step 5. Substitute \( \beta = 8 \pm 4\sqrt{3} \) back into equation (ii) to find \( \alpha \):
For \(\beta = 8 + 4\sqrt{3}\):
For \(\beta = 8 - 4\sqrt{3}\):
\(\alpha = 4(8 - 4\sqrt{3}) - 4 = 28 - 16\sqrt{3}\)
Step 6. Therefore, the possible points \( ( \alpha, \beta ) \) are:
\(( \alpha, \beta ) = (28 + 16\sqrt{3}, 8 + 4\sqrt{3}) \text{ and } (28 - 16\sqrt{3}, 8 - 4\sqrt{3})\)
Step 7. Calculate \( ( \alpha - 28 )( \beta - 8 ) \):
\(( \alpha - 28 )( \beta - 8 ) = (\pm 16\sqrt{3})(\pm 4\sqrt{3}) = 16 \cdot 4 \cdot 3 = 192\)
Thus, \(( \alpha - 28 )( \beta - 8 ) = 192\).
The Correct Answer is: 192
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Match List-I with List-II: List-I