Let P be the parabola, whose focus is (-2,1) and directrix is 2 x+y+2=0.

Question

Let P be the parabola, whose focus is $(-2,1)$ and directrix is $2 x+y+2=0$. Then the sum of the ordinates of the points on P, whose abscissa is $-$2, is

Accepted Answer

$$\begin{aligned}
&\begin{aligned}
& P M=P F \
& \Rightarrow \quad \frac{|2 x+y+2|}{\sqrt{5}}=\sqrt{(x+2)^2+(y-1)^2}
\end{aligned}\
\&\text { Now abscissa of } P \text { is }-2 \Rightarrow x=-2\
\&\begin{aligned}
& \left|\frac{y-2}{\sqrt{5}}\right|=\sqrt{0+(y-1)^2} \Rightarrow \frac{|y-2|}{\sqrt{5}}=|y-1| \
& \Rightarrow(y-2)^2=5(y-1)^2 \
& \Rightarrow 4 y^2-6 y+1=0 \Rightarrow \text { Sum of ordinates } \
& =-\left(\frac{-6}{4}\right)=\frac{3}{2}
\end{aligned}
\end{aligned}$$

Solutions · 1