Answer:
The focus point is (2 , 0) ⇒ answer D
Step-by-step explanation:
* Lets revise the equation of the parabola in standard form
- The standard form is (x - h)² = 4p(y - k)
- The focus is (h, k + p)
- The directrix is y = k - p
- If the parabola is rotated so that its vertex is (h , k) and its axis of
symmetry is parallel to the x-axis, it has an equation of
(y - k)² = 4p(x - h)
- The focus is (h + p, k)
- The directrix is x = h - p
* Lets solve the problem
∵ The equation of the parabola is y = 1/8(x² - 4x - 12)
- Lets make x² - 4x completing square
∵ √x² = x
∴ The 1st term in the bracket is x
∵ 4x ÷ 2 = 2x
∴ The product of the 1st term and the 2nd term is 2x
∵ The 1st term is x
∴ the second term = 2x ÷ x = 2
∴ The bracket is (x - 2)²
∵ (x - 2)² = (x² - 4x + 4)
∴ To complete the square add 4 to the bracket and subtract 4 out
the bracket to keep the equation as it
∴ (x² - 4x + 4) - 4 = (x - 2)² - 4
- Lets put the equation after making the completing square
∴ y = 1/8 [(x - 2)² - 4 - 12]
∴ y = 1/8 [(x - 2)² - 16] ⇒ multiply both sides by 8
∴ 8y = (x - 2)² - 16 ⇒ add 16 to both sides
∴ 8y + 16 = (x - 2)² ⇒ take from the left side 8 as a common factor
∴ 8(y + 2) = (x - 2)²
∴ The standard form of the equation of the parabola is
(x - 2)² = 8(y + 2)
∵ The standard form of the equation is (x - h)² = 4p(y - k)
∴ h = 2 , k = -2 , 4p = 8
∵ The focus is (h , k + p)
∵ h = 2
∵ 4p = 8 ⇒ divide both sides by 4
∴ p = 2
∴ The focus = (2 , -2 + 2) = (2 , 0)
* The focus point is (2 , 0)
, 
, and 
. Converting to cylindrical coordinates, using
, and the region is given in cylindrical coordinates by 
, 
, and 
.