Question:

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

For parabolas, use the properties of the vertex and directrix to form relationships between the equation's coefficients. Solve for the coefficients to find the desired result.
Updated On: Mar 18, 2025
  • \( 7 \)
  • \( 6 \)
  • \( 8 \)
  • \( 9 \)
Hide Solution
collegedunia
Verified By Collegedunia

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: \( \alpha + \beta + \gamma = 6 \).

Was this answer helpful?
0
0