Answer:
Vertex: (4,2)
Focus: (4, 2.25)
Directrix: y = 1.75
Step-by-step explanation:
The equation of the parabola is y = x² - 8x + 18
Rearranging the equation we get,
y - 2 = (x - 4)²
⇒ (x - 4)² = y - 2 ....... (1)
Therefore, the parabola has vertex at (4,2) and axis parallel to positive x -axis.
The general form of equation of a parabola vertex at (α,β) and axis parallel to positive y-axis is
(x - α)² = 4a(y - β) ....... (2)
Comparing equation (1) and equation (2), 4a = 1
⇒ a = 0.25
Now, focus will be on the axis of the given parabola i.e. x = 4 line at a distance a above the vertex
So, the coordinates of focus is (4,2 + 0.25) = (4,2.25).
Again, the directrix of the parabola will be parallel to the x-axis and at a distance from vertex 0.25 units downward.
So, the equation of the directrix will be y = (2 - 0.25) = 1.75. (Answer)
