Question

Consider the parabola with a focus at the point (4,0) and directrix y = 3.

Which two equations can be used to correctly relate the distances from the
focus and the directrix to any point (x, y) on the parabola?

Answer 1

View attached image

Explanation:

First formula is directrix-y

Second formula is the distance formula. Just plug in the (x,y) into it.

Answer 2

`\sqrt{x^(2) + y^(2)-8x +16}` and mod(y-3)

Explanation:

Step 1 :

Given the focus is at (4,0) and the directrix is y = 3. We have to find the 2 equations which relate the distance of the given focus and the given directrix to any point (x, y) on the parabola

Step 2 :

The distance between a point P(x,y) given on the parabola and the focus (4,0)

is

`\sqrt{(x-4)^(2) + (y-0)^(2) } = \sqrt{x^(2)+16-8x + y^(2) } = \sqrt{x^(2) + y^(2)-8x +16}`

Step 3 :

The distance between the point P of (x,y) and the directrix line y = 3 is

mod (y-3)

So the 2 required equations are

`\sqrt{x^(2) + y^(2)-8x +16}` and mod(y-3)