Question

Find the vertex, axis, latus rectum, and focus of the parabolas $y^2=4x+4y.$

Answer 1

$y^2=4x+4y$

$y^2-4y=4x$

$y^2-4y+4=4x+4$

${(y-2)}^2=4\left(1\right)(x+1)$

comparing with the standard equation $Y^2=4aX$

Vertex of the parabola is

$X=0,Y=0$

$\Rightarrow x+1=0,y-2=0$

$\Rightarrow x=-1,y=2$

$\therefore (-1,2)$ is vertex

Focus of the parabola is

$X=a,Y=0$

$\Rightarrow x+1=a,y-2=0$
$\Rightarrow x+1=1,y-2=0$

$\Rightarrow x=0,y=2$

$\therefore (0,2)$

Axis of the parabola is

$Y=0$

$y-2=0$

$y=2$

Latus rectum of the parabola is $4a$

Here $a=1$

$\Rightarrow L{L}^{\prime }=4$