If the equation of the parabola with vertex $V\!\left(\tfrac{3}{2},3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to
Step-by-Step Solution
Key Concept: Use the focus-directrix definition $PS = PM$: the focus $S$ is the reflection of the vertex $V$ across the directrix; squaring both distances gives the parabola equation, from which $\alpha$, $\beta$, $\gamma$ are read off.
Distance from $V(\tfrac{3}{2},3)$ to directrix $x+2y=0$ is $\tfrac{|\tfrac{3}{2}+6|}{\sqrt{5}}=\tfrac{15}{2\sqrt{5}}$. The focus is the reflection of $V$ across the directrix: $S=(3,6)$. For any point $(x,y)$ on the parabola, $PS^2=PM^2$: $$(x-3)^2+(y-6)^2=\frac{(x+2y)^2}{5}.$$ Expanding: $5(x^2-6x+9+y^2-12y+36)=(x+2y)^2$ gives $5x^2+5y^2-30x-60y+225=x^2+4y^2+4xy$, i.e., $4x^2+y^2-4xy-30x-60y+225=0$. So $\alpha=4$, $\beta=1$, $\gamma=4$, and $\alpha+\beta+\gamma=9$.
Correct Answer: 2