Question

Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = 4?

x2=−8(y−2)
x2=−2y ​
x2=−8y
x2=−2(y−2) ​

Answer 1

I just took the test and it was x^2=-8(y-2)

Explanation:

Answer 2

x2=−8(y−2)

Explanation:

Parabola is a locus of a point which moves at the same distance from a fixed point called the focus and a given line called the directrix.

Let P(x,y) be the moving point on the parabola with

focus at S(h,k)= S(0,0)

& directrix at y= 4

Now,

|PS| = √(x-h)2 + (y-k)2

|PS| = √(x-0)2 + (y-0)2

|PS| = √ x2 + y2

Let ‘d’ be the distance of the moving point P(x,y) to directrix y- 4=0

• d= |ax +by + c|/ √a2 + b2
• d= |y-4|/ √0 + 1
• d= |y-4| units.

equation of parabola is:

• |PS| = d
• √ x2 + y2 = |y-4|

Squaring on both sides, we get:

• x2 + y2 = (y-4)2
• x2 + y2 = y2 -8y + 16
• x2 = - 8y + 16
• x2 = -8 ( y - 2)

This is the required equation of the parabola with focus at (0,0) and directrix at y= 4.