If the equation of the parabola with vertex V((3)/(2), 3) and the directrix x+2 y=0 is x^2…

Question

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is $\alpha x^2+\beta y^2-\gamma x y-30 x-60 y+225=0$, then $\alpha+\beta+\gamma$ is equal to :

Accepted Answer

Equation of axis

$$\begin{aligned}
& y-3=2\left(x-\frac{3}{2}\right) \
& y-2 x=0
\end{aligned}$$

foot of directrix

$$\begin{aligned}
& \quad y-2 x=0 \
& & \quad \Rightarrow(0,0)\
& 2 y+x=0
\end{aligned}$$

$$\begin{aligned}
& \text { Focus }=(3,6) \
& \operatorname{PS}^2=P^2 \
& (x-3)^2+(y-6)^2=\left(\frac{x+2 y}{\sqrt{5}}\right)^2 \
& 4 x^2+y^2-4 x y-30 x-60 y+225=0 \
& \Rightarrow \alpha=4, \beta=1, \gamma=4 \Rightarrow \alpha+\beta+\gamma=9
\end{aligned}$$

Solutions · 1