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### Course: Precalculus (2018 edition)>Unit 2

Lesson 7: Focus and directrix of a parabola

# Equation of a parabola from focus & directrix

The equation of a parabola is derived from the focus and directrix, and then the general formula is used to solve an example.

## Want to join the conversation?

• How to remember this formula?
• Take a piece of paper and derive the formula, just like a Sal did, a couple of times. It really works for me.
• where would you learn how to graph a parabola with a diagonal directrix?
• Rotating a graph like this requires trigonometry. It takes two equations:
x' = x * cos(theta) - y * sin(theta)
y' = y * cos(theta) + x * sin(theta)

(x', y') is the coordinate of the new point (after rotation). Theta is the angle through which you have rotated, which is the angle between the origin and the directrix. Then you substitute the parabola's equation into the rotation equations:

y = k* x^2

x' = x * cos(theta) - (kx^2) * sin(theta)
y' = (kx^2) * cos(theta) + x * sin(theta)

Theta is a known value, and everything else is given in terms of x, so you can use this information to graph the parabola.

It's actually pretty easy if you're doing it on paper, though. Graph the parabola with a horizontal directrix, then place another sheet of paper over your work, draw in your coordinate grid, and plot the line you want to use for your new directrix. Now rotate the top sheet so that the new directrix is on top of the horizontal one. Then slide the top sheet along the directrix until the vertex or focus of the parabola on the bottom sheet is at the location you want your new parabola to be. Then simply trace it.
• at shouldn't it be (a-x) ^2 + (b-y)^2?
• its the same thing. a-x and x-a are negatives of each other.
square of a number is equal to the square of the negative of that number.
eg: 5^2=25
(-5)^2 is also equal to 25
• What is the main difference of parabolas from hyperbolas?
• The distinct difference is that they are generated through different methods. A parabola is created when a plane parallel to a cone's side cuts through the cone. A hyperbola results from the intersection of the plane and the cone, but with the plane at a position that is not parallel to the side of the cone.
• What if the equation of a directrix isn't as simple as y=k or y= -k? What if it's a linear equation like y=2x+3 or y=3x-10? How can we calculate k then?
• In this case, the formula becomes entirely different. The process of obtaining the equation is similar, but it is more algebraically intensive. Given the focus (h,k) and the directrix y=mx+b, the equation for a parabola is (y - mx - b)^2 / (m^2 +1) = (x - h)^2 + (y - k)^2.
Equivalently, you could put it in general form:
x^2 + 2mxy + m^2 y^2 -2[h(m^2 - 1) +mb]x -2[k(m^2 + 1)^2 -b]y + (h^2 + k^2)(m^2 + 1) - b^2 = 0
At least, I think this last one is right. The algebra got a little messy. You can check me on it, if you like.
• Can I use the general equation to compare and solve the problem?
y^2 = 4ax?
• yes,but remember the parabola drawn there has the equation x^2=4ay
• Can someone explain to me why the square root doesnt cancel out the squared terms?
• The reason is because the squared terms are added together, and not multiplied . As Johnathan said three days ago: This is "Just like (x + 1)^2 is not the same as x^2 + 1^2." What is (x+1)^2 equal to? That would be a good exercise for you. :-)
• Why is the video showing how to solve using the equation but the hints on the test is showing how to solve it using the distance formula? Its getting really confusing
• The test is showing how to solve it using the distance formula because you need the distance formula to find out the distance between the parabola and the focus. Upvote please!
• Can cylindrical mirror/catoptric drawing be related to directrix and focus intuition of parabolas?
Where directrix is our object and image is parabola/or other conic?
(http://en.wikipedia.org/wiki/Anamorphosis)