find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curv

Find a Cartesian equation for the conic section satisfying...

Key Concepts

Cartesian Equation

A Cartesian equation is an algebraic representation of a curve on the coordinate plane using x and y variables. For a parabola, the derivation of its Cartesian equation typically involves equating the distance from an arbitrary point on the curve to the focus with its distance to the directrix, leading to an equation that characterizes the curve.

Directrix

The directrix is a fixed line used in the definition of a parabola. For any point on the parabola, the distance to the directrix equals the distance to the focus. It plays a critical role in determining the shape and orientation of the parabola.

Focus

The focus is a fixed point used in the definition of a parabola. Every point on the parabola is at the same distance from this point as it is from the directrix. The position of the focus relative to the directrix determines the orientation and the 'width' of the parabola.

Parabola

A parabola is a type of conic section defined as the set of all points that are equidistant from a fixed point, known as the focus, and a fixed line, known as the directrix. This even distance property is the fundamental characteristic used to derive its equation.

Conic Section

A conic section is a curve formed by the intersection of a plane with a double-napped cone. The study of conic sections includes several types of curves, such as circles, ellipses, parabolas, and hyperbolas, each defined by unique geometric properties and equations.

Transcript

00:01 For this problem, we are given a parabola with a focus at the coordinate 2 -4 and a directrix line at x equals negative 4.

00:10 From here, we need to figure out where the vertex is, the equation of our parabola, the points that define our lattice rectum, and finally sketch out our parabola.

00:20 So, first i'm going to plot our coordinate for our focus, and this is going to be at 2 -4, and then i'm going to sketch out our directrix line, which is that x equals negative 4.

00:45 From here we can figure out the value of our, from here we can figure out the value of our vertex by finding the point that is between our focus and our directrics.

00:57 So as you can see, our x value of negative four all the way to two is going to be six units.

01:03 So our vertex is going to be halfway between these at the same y value of our focus.

01:09 So this is going to be at negative 1, 4.

01:15 And there's our vertex.

01:16 So i'm going to write that out here.

01:22 All right.

01:23 So now that we know that our focus is to the right of our vertex, we know that our parabola is going to open to the right, which we can use this information to figure out the equation of our parabola.

01:40 So we are going to be using the format of y minus k squared is equal to 4a times x.

01:53 Minus h.

01:56 So we can plug in our vertex values for h and k with h corresponding to our x value and k and k corresponding to our y value.

02:07 So we can plug these back into our equation.

02:11 And we're going to get y minus 4 squared is equal to 4 4a times x minus 4.

02:34 Minus our value of h, which is going to be negative 1.

02:37 So minus a negative 1.

02:42 Okay, so now we need to plug in a value for a.

02:46 And we know that our a value is going to be the distance between our vertex and our focus.

02:52 So from negative 1 to 2, this is going to be 3 units.

02:56 So we're going to have an a value of 3...

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