Verified Answer :

Aliwohaish12

A parabola of the form, x = (a)y² + (b)y + c has a vertex, (h, k)
where
k = -b/2a
and
h = (a)k² + (b)k + c
In your case:
k = -0/{2(3)}
k = 0
h = 3(0)²
h = 0
The vertex is: (0, 0)
Let f = the focal width
f = 1/4a
f = 1/12
The focus is at
(h + f, k)
(1/12, 0)
The equation of the directrix is:
x = h - f
x = -1/12

The vertex will be \((0,0)\) , Focus will be \((\frac{1}{12} ,0)\) , Directrix will be \(-\frac{1}{12}\) and focal width will be \((f)=\frac{1}{12}\) of the given parabola.

What is Parabola ?

Parabola is a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side.

Equation of parabola : \((x - h)^2 = 4a(y - k)\)

The coordinates of the focus are\(:\ (a, 0)\),

And the equation of the directrix,  \(x =h-f\)

We have,

\(x = 3y^2\)   \(....(i)\)

Rewrite in vertex form,

\(x=3(y-k)^2+h\)

Using Vertex form, \(x=a(y-k)^2+h\)

Compare equation \((i)\) with Vertex form,

We get,

\(a=3,\ \ h=0,\ \ k=0\)

So, Vertex of the given expression are \((0,0)\).

Now,

To find Focus i.e. ;

For this find distance \((f)\) between vertex and focus of parabola,

i.e. Using the formula, \((f)=\frac{1}{4a}\)

i.e. put value of \(a\) in \(\frac{1}{4a}\),

Distance  \((f)=\frac{1}{4*3}=\frac{1}{12}\)

Now,

Focus = \((h + f, k)\)

          \(=(0+\frac{1}{12} ,0)\)

So, Focus \(=(\frac{1}{12} ,0)\)

Now,

Find Directrix;

Using the given formula,

Directrix,  \(x =h-f\)

\(x =0-\frac{1}{12}\)

\(x =-\frac{1}{12}\)

So, Directrix of given parabola is \(-\frac{1}{12}\) .

Hence, we can say that the vertex will be \((0,0)\) , Focus will be \((\frac{1}{12} ,0)\) , Directrix will be \(-\frac{1}{12}\) and focal width will be \((f)=\frac{1}{12}\) of the given parabola.

To know more about Parabola click here

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