Identify the conic section whose equation is x^2+4 y^2-16 y+12=0 and writ the equation in standa

Identify the conic section whose equation is x^2+4 y^2-16...

Key Concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Identifying a conic section often involves analyzing the quadratic equation in two variables and examining the coefficients and terms to determine its type.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a form that reveals important features, such as vertices or centers of conic sections. This technique involves adding and subtracting terms to form a perfect square trinomial, which simplifies the equation and helps to express the conic section in its standard form.

Standard Form of an Ellipse

The standard form of an ellipse is typically given by the equation (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center of the ellipse and a and b are the semi-major and semi-minor axes. Writing an equation in this form allows for easy identification of the key characteristics of the ellipse, such as its center, axes lengths, and orientation.

Transcript

00:01 So in the following problem, we want to identify which connect section this equation represents.

00:05 We also want to shift it towards the origin and graph it.

00:09 To do so, i'm going to write it in standard form.

00:13 So we start off by taking all of our y terms on one side of the equation, and then our x and long term to the other hand side.

00:22 So we'll have that 12 minus 4x.

00:25 Now, the left hand side of the equation is an uncompleted square, so i'm just going to complete it by adding a plus 1 on this side of the equation and i'm going to add a plus 1 over here so that's just going to be 13 minus 4 x this can just be written as y minus 1 square and our right hand side stays the same now i'm going to solve for 4x so in this case i'm going to add 4 on both sides of the equation and i'm going to subtract this term right here on both sides so we'll have 13 minus y minus 1 square.

01:06 Now i'm going to divide x by 4 so we'll have x it's just equal to 13 divided by 4 because we need to do the same on this hand side of the equation it's equal to y minus 1 square divided by 4.

01:22 So this equation right here is in the form of a parabola that lies on the x -axis so it'll look something like this now we want to shift this parabola towards the origin and to do so so we're just going to rewrite it as x minus 13 over 4 is equal to negative y minus 1 square over 4.

01:48 Now i'm going to substitute the x prime to be x minus 13 over 4 and y prime is equal to y minus 1.

01:59 If we do that substitution, we'll have that x is just equal to x prime, it's equal to negative y minus 1 square over 4.

02:08 Sorry, y prime square over 4.

02:15 And we can just rewrite this as 4x is equal to negative y prime square.

02:24 So this is our answer for our translated parabola.

02:31 So now we have that the equation 4x is equal to negative y prime square, which is just this one right here...

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