Answer:
The equation of the parabola is (x - 4)² = -16(y - 1)
Explanation:
The standard form of the equation of the parabola is (x - h)² = 4p(y - k), where
- The vertex of the parabola is (h, k)
- The directrix is at y = k - p
∵ 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)