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Focus and directrix of a parabola conic sections Video transcript - [Voiceover] What I have attempted to draw here in yellow is a parabola, and as we've already seen in previous videos, a parabola can be defined as the set of all points that are equidistant to a point and a line, and the point is called the focus of the parabola, and the line is called the directrix of the parabola.
What I want to do in this video, it's gonna get a little bit of hairy algebra, but given that definition, I want to see, and given that definition, and given a focus at the point x equals a, y equals b, and a line, a directrix, at y equals k, to figure out what is the equation of that parabola actually going to be, and it's going to be based on a's, b's, and k's, so let's do that.
So let's take a arbitrary point on the parabola. Let's say we take this point right over here, and its x-coordinate is x, and its y-coordinate is y, and by definition, in order for this to be a parabola, it has to be equidistant to its focus and its directrix, so what does that mean?
That means that the distance to the directrix, which I'm drawing here in blue, has to be the same as the distance to the focus, which I am drawing in magenta, and when we take the distance to the directrix, we literally just drop a perpendicular, I guess you could say say, that is, that's going to be the shortest distance to that line, but the distance to the focus, well we see that's at a bit of an angle, and we might have to use the distance formula, which is really just the Pythagorean Theorem.
So let's do that. This distance has to be the same as that distance. So, what's this blue distance? Well, that's just gonna be our change in y. It's going to be this y, minus k. It's just this distance. So it's going to be y minus k. Now we have to be careful. The way I've just drawn it, yes, y is greater than k, so this is going to give us a positive value, and you need a non-negative value if you're talking about distances, but you can definitely have a parabola where the y-coordinate of the focus is lower than the y-coordinate of the directrix, in which case this would be negative.
So what we really want is the absolute value of this, or, we could square it, and then we could take the square root, the principle root, which would be equivalent to taking the absolute value of y minus k. So that's this distance right over here, and by the definition of a parabola, in order for x,y to be sitting on the parabola, that distance needs to be the same as the distance from x,y to a,bto the focus.
So what's that going to be? Well, we just apply the distance formula, or really, just the Pythagorean Theorem.
It's gonna be our change in x, so, x minus a, squared, plus the change in y, y minus b, squared, and the square root of that whole thing, the square root of all of that business.
Now, this right over here is an equation of a parabola. It doesn't look like it, it looks really hairy, but it IS the equation of a parabola, and to show you that, we just have to simplify this, and if you get inspired, I encourage you to try to simplify this on your own, it's just gonna be a little bit of hairy algebra, but it really is not too bad.
You're gonna get an equation for a parabola that you might recognize, and it's gonna be in terms of a general focus, a,band a gerneral directrix, y equals k, so let's do that. So the simplest thing to start here, is let's just square both sides, so we get rid of the radicals.The standard and vertex form equation of a parabola and how the equation relates to the graph of a parabola.
C - x intercepts of the graph of a quadratic function The x intercepts of the graph of a quadratic function f given by f(x) = a x 2 + b x + c are the real solutions, if they exist, of the quadratic equation a x 2 + b x + c = 0 The above equation has two real solutions and therefore the graph has x intercepts when the discriminant D = b 2 - 4 a c is positive.
This calculator will find either the equation of the parabola from the given parameters or the axis of symmetry, focus, vertex, directrix, focal parameter, x-intercepts, y-intercepts of the entered parabola. The equation of a parabola is derived from the focus and directrix, and then the general formula is used to solve an example.
In analytic geometry, the graph of any quadratic function is a parabola in the urbanagricultureinitiative.com a quadratic polynomial of the form (−) + (−) +the numbers h and k may be interpreted as the Cartesian coordinates of the vertex (or stationary point) of the urbanagricultureinitiative.com is, h is the x-coordinate of the axis of symmetry (i.e.
the axis of symmetry has equation x=h), and k is the minimum value (or. Search the world's information, including webpages, images, videos and more.
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