Question

Select the correct answer. What is the equation of the directrix of the parabola given by the equation (y − 3)2 = 8(x − 5)? A. y = 3 B. x = 3 C. x = 5 D. y = -5

Answer 1

Choice B: x = 3

Explanation:

The standard equation of a parabola is

`4p\left(x-h\right)=\left(y-k\right)^2`

where

(h, k) is the vertex and |p| is the focal length

The given equation is

`(y - 3)^2 = 8(x - 5)`

Convert this to standard form and compare with general equation

• Switch the sides of the equation
  
  `8(x - 5) = (y - 3)^2`
• 
  `8 = 2 \cdot 4`
  
  `\rightarrow 4 \cdot 2 (x - 5) = (y - 3)^2\\`
• Compare:
  
  `4 \cdot 2 (x - 5) = (y - 3)^2\\4 \cdot p(x -h) = (y - k)^2`
• This gives us
  
  `\mathrm{p = 2, vertex (h, k) = (5, 3)}`
  
• This parabola is symmetric about the x-axis and the directrix is a line parallel to the y-axis at distance -p from the x-coordinate of the vertex
  
  `x = 5-p\\\\x = 5 - 2\\\\x = 3\\\\`

Answer: Choice B: x = 3