Question

Math

Answer 1

Good evening Sarah. How far have you gotten with solving this one?

Answer 2

@Sarah10 are you there?

Answer 3

yes i am

Answer 4

Great, do you know the formula for equation of a parabola given the focus and directrix?

Answer 5

is it like (x-h)=4(y-k) something like this?

Answer 6

I don't think that is correct as written. Nevertheless, that's not the formula I prefer to use. Allow me to introduce you to a formula that is more intuitive in this case.

Answer 7

please do

Answer 8

You are given two points:
The focus, which is given by \((x_1, y_1) = (-2, 4)\)
AND
The directrix which is given by \((x_2, y_2) = (x,2)\)

In this case, the equation of the parabola can be written as:
\((x - x_1)^2 + (y - y_1)^2 = (x - x_2)^2 + (y - y_2)^2\)

The next step would be to enter the given points to the formula and then simplify from there.

Answer 9

@Sarah10 do you think you might be able to insert the given points in to the formula?

Answer 10

yes

Answer 11

Okay, would you mind typing it here using \(\LaTeX\)?

Answer 12

for the other x do i just leave it blank?

Answer 13

Allow me to color code it for you. Then you should know what to do afterwards.

Answer 14

You are given two points:
The focus, which is given by \((\color\red{x_1}, \color\green{y_1}) = (\color\red{-2}, \color\green{4})\)
AND
The directrix which is given by \((\color\orange{x_2}, \color\purple{y_2}) = (\color\orange{x},\color\purple{2})\)

In this case, the equation of the parabola can be written as:
\((x - \color\red{x_1})^2 + (y - \color\green{y_1})^2 = (x - \color\orange{x_2})^2 + (y - \color\purple{y_2})^2\)

The next step would be to enter the given points to the formula and then simplify from there.

Answer 15

im sorry im a bit confused here are we doing the distance formula?

Answer 16

Here, allow me to demonstrate how to perform the steps since it is clear you are so deeply concerned (and a little confused) about what I am trying to show you here:

Answer 17

very much appreciated

Answer 18

Once you insert the points in to the equation of parabola you will have:
\((x - \color\red{(-2)})^2 + (y - \color\green{4})^2 = (x - \color\orange{x})^2 + (y - \color\purple{2})^2\)

Answer 19

Simplifying this further we will end up with:
\((x + \color\red{2})^2 + (y - \color\green{4})^2 = (0)^2 + (y - \color\purple{2})^2\)

or simply

\((x + \color\red{2})^2 + (y - \color\green{4})^2 = (y - \color\purple{2})^2\)

Answer 20

Expanding the binomial squares yield the following:
\((x^2 + 4x + 4)+(y^2 - 8y + 16) = y^2 - 4y + 4\)

Removing the parentheses you have:
\(x^2 + 4x + 4+y^2 - 8y + 16 = y^2 - 4y + 4\)

Answer 21

Notice that we can easily subtract \( y^2\) and \(4\) from both sides which will allow us to reduce the equation to just:
\(x^2 + 4x - 8y + 16 = - 4y \)

Answer 22

Placing like terms on the same side we have:
\(x^2 + 4x + 16 = 8y - 4y\)

Which easily simplifies to just:
\(x^2 + 4x + 16 = 4y\)

Answer 23

Notice that we can rewrite the quadratic trinomial on the left side of the equation as:
\(x^2 + 4x + 4 + 12 = 4y\)

And then convert it back to a binomial square:
\((x + 2)^2 + 12 = 4y\)

Answer 24

Lastly, we can simply divide both sides by 4:
\(\dfrac{(x + 2)^2}{4} + \dfrac{12}{4} = y\)

Which simplifies to
\(\dfrac{(x + 2)^2}{4} + 3 = y\)

And can be re-written as:
\(y = \dfrac{(x + 2)^2}{4} + 3\)

Which is the equation of the parabola in most simplified form.

Answer 25

And to answer your question, yes the formula used to solve this question was derived from the distance formula.

Answer 26

Any questions @Sarah10

Answer 27

Thank you soooooo much! I've got to save this work for future references!! , thank you for taking your time with me

Answer 28

You're most welcome.