Question

How do you find the vertex, the coordinates of focus, and the directrix of the parabola \[10x = {y^2}\] ?

Answer 1

Hint: From the question, we can say that it is a concave right parabola. This is so because the question is in the form of \[{y^2} = 4ax\]. Now, we can find out the vertex, focus, and the directrix of the parabola according to the form given.

Complete step by step solution:
As we know that the parabola is a concave right parabola because of the form it is having. The form is:
\[{y^2} = 4ax\]
We will find the vertex first. We can see that here any transformations are not taking place in the parabola. Therefore, the vertex is \[(0,0)\]. Now, we will find the focus of the parabola. The form of the focus of parabola is \[(a,0)\]. From the formula \[{y^2} = 4ax\], we know that the \[a\] here is the focus of the parabola. So, we need to find the \[a\] from the question given. According to the formula we get that:
\[10x = 4ax\]

We can rewrite this equation as:
\[ \Rightarrow 4ax = 10x\]
Now, we have to shift \[4x\]to the other side, so that \[a\] is alone, and we can get the focus of the parabola.
\[ \Rightarrow a = \dfrac{{10x}}{{4x}}\]
 Now, the terms which are divisible or are similar gets cancelled here, and then we get:
\[ \Rightarrow a = \dfrac{5}{2}\]
Hence, according to the frame of the focus of the parabola, we get that the coordinates of the focus of the parabola is \[\left( {\dfrac{5}{2},0} \right)\].
Now, we are going to discover the directrix of the parabola. The equation to discover the directrix of the parabola is:
\[x = - a\]
We know that \[a = \dfrac{5}{2}\]. Therefore, \[x = - \dfrac{5}{2}\].

So, the directrix of the parabola is \[ - \dfrac{5}{2}\].

Note: In this question there were no transformations in the parabola so the answer came as \[(0,0)\]. Otherwise, the formula for vertex is \[y = a{(x - h)^2} + k\] where \[(h,k)\] are known as the vertex. This was the formula to find the vertex of Quadratic Equations.