find an x^' y^'-equation for the graph of the given x y-equation so that the center or, in the c

step 1 Welcome to this lesson. In this lesson we will look for the center of this curve then after that

Find an x^' y^'-equation for the graph of the given x...

Key Concepts

Translation of Axes

This concept involves shifting the coordinate system in order to simplify the equation of a curve by relocating its key features (such as the center or vertex) to the origin. By introducing new variables that account for the horizontal and vertical shifts, the curve's equation becomes easier to analyze and graph.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions into a perfect square plus a constant. This process is essential when converting general quadratic equations into their standard forms, as it highlights the geometric structure of the conic section, reveals the translation involved, and simplifies further analysis.

Standard Form of Conic Sections

Standard forms provide a clear and concise representation of conic sections by showing their key parameters explicitly. Once the equation is rewritten in standard form, it becomes straightforward to identify properties like the center, vertices, axes lengths, and orientation of the conic, making it easier to classify the type of curve and understand its geometry.

Classification of Conic Sections

This concept involves determining the type of conic—ellipse, parabola, or hyperbola—based on the coefficients of the quadratic equation after completing the square and translating the axes. By examining these coefficients in the standard form, one can distinguish the nature of the curve and infer its geometric properties.

Transcript

0:00 Welcome to this lesson.

00:02 In this lesson we will look for the center of this curve then after that we'll draw it.

00:11 Okay, so let's try to see what nature would become.

00:15 Now one indication for one indication looking at this is x squared plus y squared.

00:28 So this would become an ellipse.

00:34 Okay, the first thing that will do is to group the variables this the y part and that is called zero we can factor out the two so that the coefficient of the highest power of x always becomes one and this becomes four x for this one the y has a coefficient y squared has a coefficient of one so not doing nothing to that now this part would take the one that is having the x the coefficient of that we divided by two and square it so that we can know the constant part for that's so that we can we can complete the square so this is four okay it means that we would have four x plus four in other to complete that so now that i've added four in a bracket i've actually added 8 in all because this multiply over there and now this one we have y squared minus 8 y so with this one to take the negative 8 divided by 2 and square that now this becomes 16 so we know that the constant power add 16 to it so now i added 8 and 16 yes so that at the end of the day it will neutralize everything we still come back with the original equation now the whole thing here becomes x plus two squared then this becomes y minus four squared that is equal to 24 now we divide through by 24 so that we have all equals to one at our side now this becomes x plus 2 squared all over 12, all over 24 is equal to 1.

03:16 So this is an ellipse with center negative 2, 4.

03:33 We're looking at x minus x squared and y minus k squared.

03:39 So the h and the k are the centers.

03:42 Now we have the major is on parallel to the y so that's why we have the bigger side a now we can draw so there's the center we can draw the whole ellipse like this so this the y has the major is and here is two that's a negative two four so let me draw that well so with this one we have a negative 2 and 4 over there.

05:02 Negative 2 and 4...

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