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

The equation of the parabola is y = x²/4

Step-by-step explanation:

The given focus of the parabola = (0, 1)

The directrix of the parabola is y = -1

A form of the equation of a parabola is presented as follows;

(x - h)² = 4·p·(y - k)

We note that the equation of the directrix is y = k - p

The focus = (h, k + p)

Therefore, by comparison, we have;

k + p = 1...(1)

k - p = -1...(2)

h = 0...(3)

Adding equation (1) to equation (2) gives;

On the left hand side of the addition, we have;

k + p + (k - p) = k + k + p - p = 2·k

On the right hand side of the addition, we have;

1 + -1 = 0

Equating both sides, gives;

2·k = 0

∴ k = 0/2 = 0

From equation (1)

k + p = 0 + 1 = 1

∴ p = 1

Plugging in the values of the variables, 'h', 'k', and 'p' into the equation of the parabola, (x - h)² = 4·p·(y - k), gives;

(x - 0)² = 4 × 1 × (y - 0)

∴ x² = 4·y

The general form of the equation of the parabola, y = a·x² + b·x + c, is therefore;

y = x²/4.

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step-by-step explanation:

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