Derive the equation of the parabola with a focus at (0, 1) and a directrix of y = -1.

Solution:

Focus = (0, 1)

Directrix of y = - 1

Consider a point (x, y) on the parabola

The distance of a point on the parabola from directrix is same to its distance from the focus

Distance of (x, y) from the focus (0, 1) is √[(x − 0)² + (y - 1)²]------->(1)

Distance of (x, y) from the directrix y = -1 is |y + 1|-----> (2)

So the equation will be from (1) and (2)

√[(x − 0)² + (y - 1)²] = |y + 1|

Square on both sides

(x - 0)² + (y - 1)² = (y + 1)²

So we get

x² + y² - 2y + 1 = y² + 2y + 1

x² = 4y

Therefore, the equation of the parabola is x² = 4y.

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Derive the equation of the parabola with a focus at (0, 1) and a directrix of y = -1.

Summary:

The equation of the parabola with a focus at (0, 1) and a directrix of y = -1 is x² = 4y.