Find the vertex, focus, directrix, and focal width of the parabola. x = 4y²

Solution:

Given, the equation of the parabola is x = 4y².

We have to find the vertex, focus, directrix, and focal width of the parabola.

On rewriting the equation, y² = (1/4)x ------------ (1)

The general equation of the parabola is given by

(y - k)² = 4p(x - h) -------------------------- (2)

Where, (h, k) is the vertex

(h+p, k) is the focus

y = h - p is the directrix

|4p| is the focal width.

On comparing (1) and (2),

k = 0

h = 0

Therefore, vertex = (0, 0)

4px = (1/4)x

4p = 1/4

So, p = 1/16

h + p = 0 + 1/16 = 1/16

Therefore, focus = (1/16, 0)

y = 0 - 1/16

y = -1/16

Therefore, the directrix is y = -1/16

|4p| = |4(1/16)|

= 1/4

Therefore, the focal width is 1/4.

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Find the vertex, focus, directrix, and focal width of the parabola. x = 4y²

Summary:

The vertex, focus, directrix, and focal width of the parabola x = 4y² are (0, 0), (1/16, 0), y = -1/16 and 1/4.