Question

How do you find the focus, vertex, and directrix of \[{{y}^{2}}=-4x\]?

Answer 1

Hint: From the question given, we have been asked to find the focus, vertex and directrix of \[{{y}^{2}}=-4x\].We can solve the given question by using the general form of the equation in geometry concept. The given equation for a parabola is in the form of \[{{y}^{2}}=-4ax\] then its vertex is \[\left( 0,0 \right)\] , focus is \[\left( -a,0 \right)\]and directrix is \[x=a\].First of all, we have to find out in which form does the given equation is in. Then by comparing the coefficients and terms in both the general form and the given equation, we get the focus, vertex and directrix for the given question.

Complete step by step answer:
From the question given, we have been given that \[{{y}^{2}}=-4x\]
We can write the above equation as \[{{y}^{2}}=4\left( -1 \right)x\]
We can clearly observe that the given equation from the question is in the form of \[{{y}^{2}}=-4ax\] which is the general form of a parabola.
If it is so, then its vertex is \[\left( 0,0 \right)\] , focus is \[\left( -a,0 \right)\]and directrix is \[x=a\]

Now, we have to compare the terms and coefficients of both the equations to get the focus, vertex and directrix for the given equation.
By comparing both the equations, we get
Its vertex is \[\left( 0,0 \right)\], focus is \[\left( -1,0 \right)\]and directrix is \[x=1\]
Therefore, we got the focus, vertex and directrix for the given equation.

Note:
We should be very careful while comparing the given equation and general form of the equation. Also, we should be well aware of the concepts of geometry. Also, we should be well known about the terms like focus, vertex and directrix. Also, we should be very careful while writing the focus and directrix for the given equation. Similar to parabola we have curves like hyperbola, ellipse and many more. For hyperbola the general form is $\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1$ the vertices are $\left( a,0 \right)$ and $\left( -a,0 \right)$ , the focus are $\left( c,0 \right)$ and $\left( -c,0 \right)$ where the value of $c$ is $\sqrt{{{a}^{2}}+{{b}^{2}}}$ .