Now, substitute the value of a=2 into the standard equation x² = 4ay to find the specific equation for this parabola.

Find the equation of the parabola with vertex at $(0{,}0)$ and focus at $(0{,}2)$ .

Answer: The equation of the parabola is $x^2 = 8y$ .

Step-by-step solution

Step 1: Identify the orientation of the parabola

The vertex of the parabola is at the origin $(0{,}0)$ and the focus is at $(0{,}2)$ . Since the $x$ -coordinate of the vertex and focus are the same, and the $y$ -coordinate of the focus is greater than the $y$ -coordinate of the vertex, the parabola opens upwards.

Step 2: Determine the value of 'a'

For a parabola with vertex at $(0{,}0)$ and focus at $(0,a)$ , the distance from the vertex to the focus is $a$ . In this case, $a$ is the absolute difference between the $y$ -coordinates of the focus and the vertex, which is $2$ .

Step 3: Write the standard equation of the parabola

Since the parabola opens upwards and its vertex is at the origin, its standard equation is of the form $x^2 = 4ay$ .

Step 4: Substitute the value of 'a' into the equation

Now, substitute the value of $a=2$ into the standard equation $x^2 = 4ay$ to find the specific equation for this parabola.

Step 5: Simplify the equation

Perform the multiplication to simplify the equation and obtain the final equation of the parabola.

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Find the equation of the parabola with vertex at (0,0)(0{,}0)(0,0) and focus at (0,2)(0{,}2)(0,2).