Write the equation of a parabola with focus and directrix. 

Solution:

We will use the concept of the parabola to find its equation.

Let the focus of the parabola be (a, b) and its directrix be y = c.

Suppose (\(x_0, y_0\)) is any point on this parabola.

We will use the distance formula to write the distance between (a, b) and (\(x_0, y_0\)).

Distance between (a, b) and (\(x_0, y_0\)) = √ (\(x_0\) - a)² + (\(y_0\) - b)²

Distance between y = c and (\(x_0, y_0\)) = |\(y_0\) - c|

Both these distances are equal.

So, √ (\(x_0\) - a)² + (\(y_0\) - b)² = |\(y_0\) - c|

On squaring both sides, we get (\(x_0\) - a)² + (\(y_0\) - b)² = (\(y_0\) - c)²

(\(x_0\) - a)² + b² - c² = 2 (b - c) \(y_0\)

Since (\(x_0, y_0\)) is any arbitrary point on this parabola, the equation of the parabola will be (x - a)² + b² - c² = 2 (b - c) y

Write the equation of a parabola with focus and directrix.

Summary:

(x - a)² + b² - c² = 2 (b - c) y, where (a, b) is the focus of the parabola and its directrix is y = c.