Question

How do you find the vertex, focus and directrix of $y - 2 = \left( { - \dfrac{1}{8}} \right){\left( {x + 2} \right)^2}$?

Answer 1

Hint: First, we must find the curve which resembles the given equation by comparing it to the general equations of various conic sections. Then, we should compare the two equations in order to determine the vertex and foci of the given equation, and also to determine their orientation. Then we will use the formula of directrix to find the directrix of the conic section represented by the given equation. We should also know that if the origin is shifted to $\left( {h,k} \right)$, then X becomes $\left( {x - h} \right)$ and Y becomes $\left( {y - k} \right)$.

Complete answer:
Given, $y - 2 = \left( { - \dfrac{1}{8}} \right){\left( {x + 2} \right)^2}$
$ \Rightarrow {\left( {x + 2} \right)^2} = - 8\left( {y - 2} \right) - - - - \left( 1 \right)$
Here, in this equation, one variable is of degree $2$ and another variable is of degree $1$.
So, this is a parabola.
The degree of the term containing x is $2$ and y is $1$. So, the parabola must be in vertical orientation. Since the coefficient of the term consisting of y is negative, the parabola must be facing downwards.
General form of a vertical parabola facing downwards is ${\left( {x - h} \right)^2} = \left( { - 4a} \right)\left( {y - k} \right)$.
Comparing equation $(1)$ with the general form of equation, we get,
$h = - 2$
$k = 2$
$ - 4a = - 8$
So, the coordinates of the vertex of the parabola are: $\left( {h,k} \right)$.
So, the coordinates of the vertex of the given parabola are: $\left( { - 2,2} \right)$.
Now, we also have $ - 4a = - 8$.
Dividing both sides of the equation by $ - 4$.
$ \Rightarrow a = 2$
So, the value of a is $2$.
Now, for a parabola facing downwards having vertex at origin, the directrix has an equation $y = a$.
But the parabola given to us has a vertex at $\left( { - 2,2} \right)$.
So, we use the concept of shifting of origin to find the directrix of this parabola.
So, we have, equation of directrix as: $Y = a$
Putting in the value of a, we get,
$ \Rightarrow Y = 2$
Now, we replace Y with $\left( {y - 2} \right)$ to find the equation of the directrix for this shifted parabola. So, we get,
$ \Rightarrow y - 2 = 2$
$ \Rightarrow y = 4$
Hence, the equation of the directrix of the parabola is $y = 4$.
Similarly, the coordinates of focus of a downward facing parabola are $\left( {0, - a} \right)$.
But the parabola given to us has a vertex at $\left( { - 2,2} \right)$.
So, we use the concept of shifting of origin to find the coordinates of focus.
So, we have, coordinates of focus as: $X = 0$ and $Y = - a$
Now, we replace the X with $\left( {x + 2} \right)$ and Y with $\left( {y - 2} \right)$ to find the coordinates of the focus of the parabola.
So, we get,
$ \Rightarrow x + 2 = 0$ and $y - 2 = - a$
Putting in the value of a, we get,
$ \Rightarrow x = - 2$ and $y - 2 = - 2$
$ \Rightarrow x = - 2$ and $y = 0$
So, the coordinates of focus of parabola are: $\left( { - 2,0} \right)$.

Note:
For a parabola, the variable with degree $2$ gives us the idea, whether the parabola is symmetric to $x - axis$ or $y - axis$. Then, depending upon the sign of $a$, we can determine the orientation of the parabola. Like, for a parabola symmetric to $x - axis$, if $a$ is positive, then the parabola is right handed, and if $a$ is negative, then the parabola is left handed. And, for a parabola symmetric to $y - axis$, if$a$is positive, then the parabola is open upwards, and if $a$ is negative, then the parabola is open downwards. In this way, we can determine the orientation of the parabola. We must know the concept of shifting of origin in order to solve the given question.