If the equation of the parabola with vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) is \[ ax^2 + b y^2 - cxy - 30x - 60y + 225 = 0, \text{ then } \alpha + \beta + \gamma \text{ is equal to:} \]
If the equation of the parabola with vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) is \[ ax^2 + b y^2 - cxy - 30x - 60y + 225 = 0, \text{ then } \alpha + \beta + \gamma \text{ is equal to:} \]
Show Hint
- \( 7 \)
- \( 6 \)
- \( 8 \)
- \( 9 \)
The Correct Option is B
Solution and Explanation
The equation of the parabola is given in general form. We use the condition of the vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) to derive the values of \( a \), \( b \), and \( c \).
Then, we calculate \( \alpha + \beta + \gamma \).
Final Answer: \( \alpha + \beta + \gamma = 6 \).
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