A parabola can be drawn given a focus of (4, -3) and a directrix of y=5 Write the equation of the parabola in any form.

The equation of the parabola is (x - 4)² = -16(y - 1)

Step-by-step explanation:

The standard form of the equation of the parabola is (x - h)² = 4p(y - k), where

∵ The focus of the parabola is (4, -3)

∵ The focus is (h, k + p)

∴ h = 4

∴ k + p = -3 ⇒ (1)

∵ It has a directrix of y = 5

∵ The directrix of the parabola is y = k - p

∴ k - p = 5 ⇒ (2)

→ Add equations (1) and (2) to find k and p

∵ (k + k) + (p - p) = (-3 + 5)

∴ 2k + 0 = 2

∴ 2k = 2

→ Divide both sides by 2

∴ k = 1

→ Substitute the value of k in equation (1)

∵ 1 + p = -3

→ Subtract 1 from both sides

∴ 1 - 1 + p = -3 - 1

∴ p = -4

∵ The form of the equation of the parabola is (x - h)² = 4p(y - k)

→ Substitute the values of h, k, p in it

∴ (x - 4)² = 4(-4)(y - 1)

∴ (x - 4)² = -16(y - 1)

∴ The equation of the parabola is (x - 4)² = -16(y - 1)