MATH SOLVE

2 months ago

Q:
# I WILL GIVE BRAINLIEST Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = 2?a. x2=−4yb. x2=−4(y−1)c. x2=−(y−1)d. x2=−y

Accepted Solution

A:

The correct option is: (B) x2=−4(y−1)

Explanation:

The distance between (x,y) and the focus point is: [tex] \sqrt{(x-x_o)^2 + (y-y_o)^2} [/tex] (Distance formula)

[tex]x_o = 0 \\ y_o = 0[/tex]

Plug in the values; the expression becomes: [tex] \sqrt{x^2 + y^2} [/tex]

The distance between (x,y) and the directrix y = 2: [tex] \sqrt{(y-2)^2} [/tex]

In parabola both distances are equal; therefore

[tex] \sqrt{x^2 + y^2} [/tex] = [tex] \sqrt{(y-2)^2} [/tex]

Take square on both sides:

[tex]x^2 + y^2 = (y-2)^2 \\ x^2 + y^2 = y^2 + 4 - 4y \\ x^2 = 4-4y \\ x^2 = -4(y-1) \\[/tex]

Hence the correct option is: (B) x2=−4(y−1)

Explanation:

The distance between (x,y) and the focus point is: [tex] \sqrt{(x-x_o)^2 + (y-y_o)^2} [/tex] (Distance formula)

[tex]x_o = 0 \\ y_o = 0[/tex]

Plug in the values; the expression becomes: [tex] \sqrt{x^2 + y^2} [/tex]

The distance between (x,y) and the directrix y = 2: [tex] \sqrt{(y-2)^2} [/tex]

In parabola both distances are equal; therefore

[tex] \sqrt{x^2 + y^2} [/tex] = [tex] \sqrt{(y-2)^2} [/tex]

Take square on both sides:

[tex]x^2 + y^2 = (y-2)^2 \\ x^2 + y^2 = y^2 + 4 - 4y \\ x^2 = 4-4y \\ x^2 = -4(y-1) \\[/tex]

Hence the correct option is: (B) x2=−4(y−1)