Q.The directrix of a parabola is x+y+4=0 and vertex is at (-1,-1). Find the position of the focus and the equation of parabola.

The vertex is the midpoint of the focus and the foot of the perpendicular from the vertex to the directrix; use this to locate the focus, then apply the focus–directrix distance definition to get the equation.

Find the foot of perpendicular ($M$) from vertex $V(-1,-1)$ to the directrix $x+y+4=0$: using the foot-of-perpendicular formula for point $(x_0,y_0)$ to line $ax+by+c=0$:

$M=(x_0,y_0)-\dfrac{ax_0+by_0+c}{a^2+b^2}(a,b).$

Here $a=b=1,c=4,(x_0,y_0)=(-1,-1)$: $ax_0+by_0+c=-1-1+4=2$, $a^2+b^2=2$.

$M=(-1,-1)-\dfrac22(1,1)=(-1,-1)-(1,1)=(-2,-2).$

Find the focus $F$: since the vertex is the midpoint of $F$ and $M$: $V=\dfrac{F+M}2\implies F=2V-M=2(-1,-1)-(-2,-2)=(-2,-2)-(-2,-2)=(0,0)$.

…