Question

28. What are the vertex, focus, and directrix of the parabola with equation y = x^2 - 8x + 18? (1 point)

A)vertex: (2,4)
focus: (2, 2.25)
directrix: y = 1.75
vertex: (4,2)
focus: (4, 2.25)
directrix: y = 1.75
vertex: (4, -2)
focus: (4, -1.75)
directrix: y = 1.75
vertex: (4,2)
focus: (4, 1.75)
directrix: y = 2.25

Answer 1

Vertex: (4,2)

Focus: (4, 2.25)

Directrix: y = 1.75

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)