If a parabola has the focus at (2,4)\left(2,4\right)(2,4) and a directrix line x=−4x=-4x=−4 , find the standard equation for the parabola.

12 ( x + 1 ) = ( y − 4 ) 2 12\left(x+1\right)=\left(y-4\right)^2 12 ( x + 1 ) = ( y − 4 ) 2

If a parabola has the focus at ( 2 , 4 ) \left(2,4\right) ( 2 , 4 ) and a directrix line x = − 4 x=-4 x = − 4 , find the standard equation for the parabola.

Hey everyone. Let's see if we can do this problem. So in this problem we are told if a parabola has the focus point at two four, and a directrix line at x equals negative four, find the standard equation for the parabola. Now what I'm going to do to solve this is I'm first going to draw out a graph, and we don't actually have to graph this parabola, but this will kind of give us some reference as to what we're dealing with here. So we'll have a standard x and y graph, and what I'm going to do is draw where about the focus and directrix would be. Well, you can see the focus is at the point two four, so that means horizontally we're one, two units to the right, and vertically we're up four units. So one, two, three, four, this would be our focus. So if this is the focus point right here, we're told the directrix line is at x equals negative four. So that means we need to go back one, two, three, four units to find our directrix line, which would look something like that. So this is what we would have, and this would be our directrix line, and we would say that this is x equals negative four. And we know that our focus point is at two four, and so our parabola is going to be somewhere in the middle here. And if I look, this is actually the the center point of our parabola. Because notice it takes one, two, three units to get there, and it takes one, two, three units to get there. So this is right in the middle of our directrix and focus. Now the question also is gonna become what direction the parabola opens. And I can tell it's gonna open to the right because that's where our focus is, and our parabola and focus always open or go in the same direction. So since our parabola opens this way, that's why the focus would be there, or we saw the focus was already there, so our parabola must open that way. Now the standard equation for a sideways parabola like this is four p times x minus h is equal to y minus k squared. And what I can do is figure out that our vertex here, well our vertex is at a horizontal position of negative one, and a vertical position of four. And so because this is our vertex, that tells us what the H and K are going to be. Our H is going to be negative one and our K is going to be four. Now the last thing I need to actually solve this, so we only have X's and Y's left in the equation, is to figure out what this p value is. And recall that p is the or the absolute value of p is the distance to both the to both the focus point as well as the directrix. And since I can see that we have to go one, two, three units to get to the focus point, and one, two, three units to get to the directrix, the absolute value of p is three. And the p value is also just gonna be positive three, because we can know that this opens to the right, so we don't have to worry about any negative numbers. So what that means is that our p value is going to be three. So plugging everything into this equation, we're gonna have four times three times x minus negative one, and then this is going to equal y minus four squared, and really I should have bigger brackets here. So this is what we have, and four times three that's 12, and then we can cancel the negatives here to give us x plus one, and then this is equal to y minus four squared. So So this is what the equation would look like given the focus and directrix. So notice how we were able to take the focus and directrix already given to us and construct the standard equation for the parabola that we are dealing with. And this is how we can solve these types of problems when we're not at the origin and we have a sideways parabola. So hope you found this helpful. Thanks for watching, and let me know if you have any questions.

If a parabola has the focus at ( 2 , 4 ) \left(2,4\right) ( 2 , 4 ) and a directrix line x = − 4 x=-4 x = − 4 , find the standard equation for the parabola.

12 ( x + 1 ) = ( y − 4 ) 2 12\left(x+1\right)=\left(y-4\right)^2 12 ( x + 1 ) = ( y − 4 ) 2

− ( x + 1 ) = ( y − 4 ) 2 -\left(x+1\right)=\left(y-4\right)^2 − ( x + 1 ) = ( y − 4 ) 2

4 ( x − 1 ) = ( y + 4 ) 2 4\left(x-1\right)=\left(y+4\right)^2 4 ( x − 1 ) = ( y + 4 ) 2

16. Parametric Equations & Polar Coordinates

Conic Sections

Calculus

16. Parametric Equations & Polar Coordinates

Conic Sections

Struggling with Calculus?

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Multiple Choice

If a parabola has the focus at $\left(2,4\right)$ and a directrix line $x=-4$ , find the standard equation for the parabola.

$12\left(x+1\right)=\left(y-4\right)^2$

$-\left(x+1\right)=\left(y-4\right)^2$

$12x=y^2$

$4\left(x-1\right)=\left(y+4\right)^2$

Verified step by step guidance

Step 1: Understand the problem. The parabola is defined by its focus at (2, 4) and its directrix at x = -4. The equation of a parabola is derived using the definition that the distance from any point on the parabola to the focus is equal to its distance to the directrix.

Step 2: Determine the vertex of the parabola. The vertex lies midway between the focus and the directrix. The x-coordinate of the vertex is the average of the x-coordinates of the focus and the directrix: (2 + (-4))/2 = -1. The y-coordinate remains the same as the focus, which is 4. Thus, the vertex is (-1, 4).

Step 3: Identify the orientation of the parabola. Since the directrix is vertical (x = -4), the parabola opens horizontally. Specifically, it opens to the right because the focus is to the right of the directrix.

Step 4: Use the standard form of a horizontally opening parabola. The equation is (y - k)^2 = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus. Here, h = -1, k = 4, and p = 3 (distance between the vertex and focus). Substitute these values into the equation.

Step 5: Simplify the equation. Substituting h = -1, k = 4, and p = 3 into the formula gives (y - 4)^2 = 12(x + 1). This is the standard equation of the parabola.

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