What is the equation of a parabola with (−2, 4) as its focus and y = 6 as its directrix?

Guest Jun 15, 2017

#2**+1 **

There's an excellent formula that I concocted 5 days ago to always give the equation of a parabola when only the directrix and focus is given. Here is the equation:

Let a = x-coordinate of focus

Let b = y-coordinate of focus

Let k= equation of line of the directrix

\(y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}\)

All you have to do is plug into this formula, simplify, and you're done. Let's try this together:

a=-2

b=4

k=6

\(y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}\) | Plug in the appropriate values that are given by the focus and directrix. |

\(y=\frac{(x-(-2))^2}{2(4-6)}+\frac{4+6}{2}\) | Let's clean this up a bit, shall we? |

\(y=\frac{(x+2)^2}{-4}+5\) | Technicaly, you could stop here and call it a day, but I am going to attempt to make it look even cleaner! I'll expand the \((x+2)^2\) |

\(y=-\frac{x^2+4x+4}{4}+5\) | After expanding it, I'll change 5 into an improper fraction that I can add to the current fraction. |

\(y=-\frac{x^2+4x+4}{4}+\frac{20}{4}\) | Since the fractions have common denominators. Add the fractions together, but you have to be very attentive to how you do this. Notice how there is a negative. I'm going to get rid of this because you can't combine a negative fraction with a positive one; it just doesn't work. |

\(y=\frac{-(x^2+4x+4)}{4}+\frac{20}{4}\) | I'm distributing that negative because you cannot combine a negative and positive fraction. |

\(y=\frac{-x^2-4x-4}{4}+\frac{20}{4}\) | Now, you can add the fractions together. Now you add the fractions together normally. |

\(y=\frac{-x^2-4x+16}{4}\) | Break off the two last terms from the current fraction. You'll see why. |

\(y=\frac{-x^2}{4}+\frac{-4x+16}{4}\) | The rightmost fraction can be simplified because a factor of 4 goes into both terms. Wow! |

\(y=\frac{-x^2}{4}-x+4\) | |

\(y=-\frac{1}{4}x^2-x+4\) | This is your final equation. |

No, I did not just pull that formula out of the air! I derived it myself. If you are wondering why this formula will work 100% of the time, I explained it, in detail, here where someone asked a similar question to yours.

TheXSquaredFactor
Jun 15, 2017

#1**+1 **

We want the form

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

Where the vertex is (-2 ,6+ p) and p = ( y coordinate of the focus - y value of the directrix) / 2 =( 4 - 6)/2 = -2/2 = -1

So the vertex = (-2, 5) = (h,k)

So we have

4(-1) (y -5) = ( x - -2)^2 simplify

-4(y - 5) = ( x + 2)^2

-4y + 20 = x^2 + 4x + 4 subtract 20 from both sides

-4y = x^2 + 4x - 16 divide through by -4

y = (-1/4)x^2 - x + 4

Here's the graph : https://www.desmos.com/calculator/1khnfskw3p

CPhill
Jun 15, 2017

#2**+1 **

Best Answer

There's an excellent formula that I concocted 5 days ago to always give the equation of a parabola when only the directrix and focus is given. Here is the equation:

Let a = x-coordinate of focus

Let b = y-coordinate of focus

Let k= equation of line of the directrix

\(y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}\)

All you have to do is plug into this formula, simplify, and you're done. Let's try this together:

a=-2

b=4

k=6

\(y=\frac{(x-a)^2}{2(b-k)}+\frac{b+k}{2}\) | Plug in the appropriate values that are given by the focus and directrix. |

\(y=\frac{(x-(-2))^2}{2(4-6)}+\frac{4+6}{2}\) | Let's clean this up a bit, shall we? |

\(y=\frac{(x+2)^2}{-4}+5\) | Technicaly, you could stop here and call it a day, but I am going to attempt to make it look even cleaner! I'll expand the \((x+2)^2\) |

\(y=-\frac{x^2+4x+4}{4}+5\) | After expanding it, I'll change 5 into an improper fraction that I can add to the current fraction. |

\(y=-\frac{x^2+4x+4}{4}+\frac{20}{4}\) | Since the fractions have common denominators. Add the fractions together, but you have to be very attentive to how you do this. Notice how there is a negative. I'm going to get rid of this because you can't combine a negative fraction with a positive one; it just doesn't work. |

\(y=\frac{-(x^2+4x+4)}{4}+\frac{20}{4}\) | I'm distributing that negative because you cannot combine a negative and positive fraction. |

\(y=\frac{-x^2-4x-4}{4}+\frac{20}{4}\) | Now, you can add the fractions together. Now you add the fractions together normally. |

\(y=\frac{-x^2-4x+16}{4}\) | Break off the two last terms from the current fraction. You'll see why. |

\(y=\frac{-x^2}{4}+\frac{-4x+16}{4}\) | The rightmost fraction can be simplified because a factor of 4 goes into both terms. Wow! |

\(y=\frac{-x^2}{4}-x+4\) | |

\(y=-\frac{1}{4}x^2-x+4\) | This is your final equation. |

No, I did not just pull that formula out of the air! I derived it myself. If you are wondering why this formula will work 100% of the time, I explained it, in detail, here where someone asked a similar question to yours.

TheXSquaredFactor
Jun 15, 2017