Question

Find the vertex, focus and directrix of the parabolic equation $y = - \dfrac{1}{6}\left( {{x^2} + 4x - 2} \right)$ ?

Answer 1

Hint: In this question, we are going to find the vertex, focus and directrix of the parabola for the given equation. The given question contains a parabolic equation and we have to compare it to the standard form of parabola. By comparing those we will get the value of vertex, focus and directrix of the parabola. Hence, we can get the required result.

Complete step by step answer:
Here in this question, we are supposed to find the vertex, focus and directrix for the given parabolic equation.Given equation is $y = - \dfrac{1}{6}\left( {{x^2} + 4x - 2} \right)$ and it can be rewritten as,
$y = - \dfrac{1}{6}\left( {{x^2} + 4x - 2} \right) \\
\Rightarrow y = - \dfrac{1}{6}\left( {{x^2} + 4x} \right) + \dfrac{1}{3} \\
\Rightarrow y = - \dfrac{1}{6}\left( {{x^2} + 4x + 4} \right) + \dfrac{4}{6} + \dfrac{1}{3} \\
\Rightarrow y = - \dfrac{1}{6}{\left( {x + 2} \right)^2} + 1 \\ $
We know that the standard form for a parabola facing down is $y = a{\left( {x - h} \right)^2} + k$ and the vertex is $\left( {h,k} \right)$.On comparing the given parabolic equation and the standard form of the parabola, we get
$h = - 2 \\
\Rightarrow k = 1 \\
\Rightarrow a = - \dfrac{1}{6} \\ $
So, the vertex is $\left( { - 2,1} \right)$ since $a < 0$, meaning the parabola opens down.
We also know that the distance of directrix from vertex can be calculated by $d = \dfrac{1}{{4|a|}}$.
$d = \dfrac{1}{{4\left( {\dfrac{1}{6}} \right)}} \\
\Rightarrow d = \dfrac{3}{2} \\
\Rightarrow d = 1.5 \\ $
Therefore, the directrix is at
$y = 1 + 1.5 \\
\Rightarrow y = 2.5 \\ $
Directrix is at $y = 2.5$
Focus is at $\left( {h,\left( {k - d} \right)} \right)$
$\left( { - 2,\left( {1 - 1.5} \right)} \right) \\
\Rightarrow \left( { - 2, - 0.5} \right)$
Focus is at $\left( { - 2, - 0.5} \right)$.

Hence, the vertex is $\left( { - 2,1} \right)$, the focus is $\left( { - 2, - 0.5} \right)$ and the directrix is $y = 2.5$.

Note:The given question was an easy one. Students should be aware of the parabolas and related concepts. A curve where any point is at an equal distance from a fixed point and a fixed straight line is known as a parabola. Students should be extra careful while comparing the given equation with the standard form of equation.