Answer:
a. The coordinates of the vertex are (2 , 3)
b. The focal length of the parabola is 8
c. The equation, in standard form, of the parabola is (y - 3)² = -8(x - 2)²
Explanation:
* Lets revise the equation of a parabola
- If the equation is in the standard form (y − k)² = 4p(x − h), then:
# Use the given equation to identify h and k for the vertex, (h , k)
# Use the value of k to determine the axis of symmetry, y = k
# Use h , k and p to find the coordinates of the focus, (h + p , k)
# Use h and p to find the equation of the directrix, x = h − p
# The endpoints of the focal diameter, (h + p , k ± 2p)
# The focal length is I4pI
* Now lets solve the problem
∵ The parabola has focus at (0 , 3)
∵ The coordinates of its focus, (h + p , k)
∴ h + p = 0 ⇒ (1)
∴ k = 3
∵ Its directrix at x = 4
∵ The equation of the directrix, x = h − p
∴ h - p = 4 ⇒ (2)
- Add (1) and (2) to find h and p
∴ h + p + h - p = 0 + 4 ⇒ add the like terms
∴ 2h = 4 ⇒ divide both sides by 2
∴ h = 2
- Substitute the value of h in (1)
∴ 2 + p = 0 ⇒ subtract 2 from both sides
∴ p = -2
a. ∵ h = 2 and k = 3
∵ The vertex of the parabola is (h , k)
∴ The coordinates of the vertex are (2 , 3)
b. ∵ The focal length is I4pI
∵ p = -2
∴ The focal length = I4 × -2I = I-8I = 8
∴ The focal length of the parabola is 8
c. ∵ The equation in the standard form is (y − k)² = 4p(x − h)
∵ h = 2 , k = 3 , p = -2
∴ The equation is (y - 3)² = 4(-2)(x - 2)²
∴ The equation is (y - 3)² = -8(x - 2)²
∴ The equation, in standard form, of the parabola is (y - 3)² = -8(x - 2)²