Find the vertex, focus and directrix of the parabola `x^(2)=2(2x+y)`.

Answer

`"Vertex"-=(2,-2),"Focus"-=(2,-3//2),"Directrix":y=-(5)/(2)`

Explanation

We have `x^(2)=2(2x+y)`
`:." "x^(2)-4x+4=2(y+2)`
`rArr" "(x-2)^(2)=2(y+2)`
Comparing with `(x-h)^(2)=2(y-k)`, we get
Vertex of the parabola is (2,-2).
Length of latus rectum =4a=2
Axis of the parabola is x=2.
Also, parabola is concave upward.
Focus lies at distance a units above the vertex on the axis.
So, focus is `(2,-2+(1)/(2))-=(2,-(3)/(2))`.
Directrix is at distance a units below the vertex and perpendicular to the axis.
Therefore, directrix is `y=-2-(1)/(2)ory=-(5)/(2)`.