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How do you find the vertex, focus and directrix of the parabola ${\left( {x + \dfrac{1}{2}} \right)^2} = 4\left( {y - 3} \right)?$

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Last updated date: 16th Jun 2024
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Answer
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Hint: In this question, we are going to find the vertex, focus and directrix of the parabola for the given equation.
The given equation is of the form of a parabola and now we are going to compare the given values to the standard form of a parabola.
By comparing those we get the value of vertex, focus and directrix of the parabola.
Hence, we can get the required result.

Formula used: If the parabola has a horizontal axis, the standard form of the equation of the parabola is
${(y - k)^2} = 4p(x - h)$, where $p \ne 0$
The vertex of this parabola is at $\left( {h,k} \right)$.
The focus is at $\left( {h + p,k} \right)$
The directrix is the line $x = h - p$

Complete Step by Step Solution:
In this question, we are going to find the vertex, focus and directrix for the given parabolic equation.
First write the given equation and mark it as $\left( 1 \right)$
$ \Rightarrow {\left( {x + \dfrac{1}{2}} \right)^2} = 4\left( {y - 3} \right)...\left( 1 \right)$
The given equation is of the parabolic form
We compare this equation to the standard form of the parabolic equation.
Here $p = 2$
The vertex V of the parabola is $\left( {\dfrac{{ - 1}}{2},3} \right)$
The focus of the parabola is $\left( {\dfrac{{ - 1}}{2},4} \right)$
The directrix of the parabola is $\left( {3 - 1} \right) = 2$

Thus the vertex, focus and directrix of the parabola are $\left( {\dfrac{{ - 1}}{2},3} \right)$, $\left( {\dfrac{{ - 1}}{2},4} \right)$ and $2$ respectively.

Note: The parabola has two real foci situated on its axis one of which is the focus S and the other lies at infinity. The corresponding directrix is also at infinity.
A parabola is a curve where any point is at an equal distance from
A fixed point and a fixed straight line