Question

What will be the coordinates of the focus and the equation of directrix of translated parabola $x=y^2+2y+5?$

Answer 1

Given, the equation of translated parabola as
$x=y^2+2y+5.$
On rearranging the equation, we get:
$(x-4)=(y+1{)}^2$
$\Rightarrow (y+1{)}^2=4\times \frac{1}{4}(x-4)$
Comparing with the equation of translated parabola, we get the vertex of parabola as $(4,-1)$
And $a=\frac{1}{4}.$
Also, the coordinates of focus of translated parabola is given by $(a+h,k)$ i.e $(\frac{17}{4},-1).$
The equation of directrix is given by $x=-a+h$ i.e $x=\frac{15}{4}.$