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Intro

Completing the Square is the method used to transform equations of conic sections from the general equation, \(Ax^2+Cy2+Dx+Ey+F=0\), to the standard form. The notes show the different standard forms for each conic section. To learn more about each conic section, visit the following pages, Circles, Ellipses, Parabolas, Hyperbolas.

Notes

Practice Problems

Complete the square to write each equation in standard form. Then identify the conic section.

\(\textbf{1)}\) \( 9x^2+4y^2-8y=32 \)

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The equation is \( \displaystyle\frac{x^2}{4}+\displaystyle\frac{(y-1)^2}{9}=1, \) Ellipse
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\(\,\,\,\,\,\,9x^2+4y^2-8y=32\)

\(\,\,\,\,\,\,9\left(x\right)^2+4\left(y^2-2y+\text{___}\right)=32+4\left(\text{___}\right)\)

\(\,\,\,\,\,\,9\left(x\right)^2+4\left(y^2-2y+\underline{ 1 }\right)=32+4\left(\underline{ 1 }\right)\)

\(\,\,\,\,\,\,9\left(x\right)^2+4\left(y-1\right)^2=36\)

\(\,\,\,\,\,\,\displaystyle\frac{9\left(x\right)^2}{36}+\frac{4\left(y-1\right)^2}{36}=\frac{36}{36}\)

\(\,\,\,\,\,\,\displaystyle\frac{x^2}{4}+\frac{\left(y-1\right)^2}{9}=1\)

\(\textbf{2)}\) \( 2x^2-12x-y+22=0 \)

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The equation is \( y=2(x-3)^2+4, \) Parabola
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\(\textbf{3)}\) \( x^2+8x+y^2-6y=11. \)

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The equation is \( (x+4)^2+(y-3)^2=36,\) Circle
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\(\textbf{4)}\) \( x^2+4x=25y^2+250y+646 \)

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The equation is \( \displaystyle\frac{(x+2)^2}{25}-\displaystyle\frac{(y+5)^2}{1}=1, \) Hyperbola
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Show Work

\(\text{Step 1: Set up the blank spaces.}\)

\(\,\,\,\,\,\,x^2+4x=25y^2+250y+646\)

\(\,\,\,\,\,\,x^2+4x-25y^2-250y=646\)

\(\,\,\,\,\,\,\left(x^2+4x+\text{___}\right)-25\left(y^2+10y+\text{___}\right)=646 + \text{___} -25\left( \text{___} \right)\)


\(\text{Step 2: Plug } \left(\frac{b}{2}\right)^2 \text{ into the spaces.}\)

\(\,\,\,\,\,\,\left(x^2+4x+\underline{4}\right)-25\left(y^2+10y+\underline{25}\right)=646 + \underline{4} -25\left( \underline{25} \right)\)

\(\,\,\,\,\,\,\left(x^2+4x+4\right)-25\left(y^2+10y+25\right)=646 + 4 -625\)



\(\text{Step 3: Complete the square.}\)

\(\,\,\,\,\,\,(x+2)^2-25\left(y+5\right)^2=25\)


\(\text{Step 4: Divide both sides by 25 to set equal to 1}\)

\(\,\,\,\,\,\,\displaystyle\frac{(x+2)^2}{25}-\frac{25\left(y+5\right)^2}{25}=\frac{25}{25}\)

\(\,\,\,\,\,\,\displaystyle\frac{(x+2)^2}{25}-\frac{\left(y+5\right)^2}{1}=1\)

Without completing the square, identify each equation as a parabola, circle, ellipse or hyperbola.

\(\textbf{5)}\) \( 5x^2-4y^2+3x-10y=100 \)

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The answer is Hyperbola

\(\textbf{6)}\) \( 4x^2-3x+25y^2+12y-110=0 \)

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The answer is Ellipse

\(\textbf{7)}\) \( 6x^2+5x-2y-35=32 \)

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The answer is Parabola

\(\textbf{8)}\) \( 4x^2+12x+4y^2-3y=41 \)

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The answer is Circle

\(\textbf{9)}\) \( x^2=9y^2+81 \)

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The answer is Hyperbola

See Related Pages\(\)

\(\bullet\text{ All Conic Section Notes}\)
\(\,\,\,\,\,\,\,\,\)

\(\bullet\text{ Equation of a Circle}\)
\(\,\,\,\,\,\,\,\,(x-h)^2+(y-k)^2=r^2…\)

\(\bullet\text{ Parabolas}\)
\(\,\,\,\,\,\,\,\,y=a(x-h)^2+k…\)

\(\bullet\text{ Axis of Symmetry}\)
\(\,\,\,\,\,\,\,\,x=-\frac{b}{2a}…\)

\(\bullet\text{ Ellipses}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1…\)

\(\bullet\text{ Area of Ellipses}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\pi a b…\)

\(\bullet\text{ Hyperbolas}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1…\)

\(\bullet\text{ Conic Sections- Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+8x+y^2−6y=11 \Rightarrow (x+4)^2+(y−3)^2=36…\)

\(\bullet\text{ Conic Sections- Parametric Equations}\)
\(\,\,\,\,\,\,\,\,x=h+r \cos{t}\)
\(\,\,\,\,\,\,\,\,y=k+r \sin{t}…\)

\(\bullet\text{ Degenerate Conics}\)
\(\,\,\,\,\,\,\,\,x^2−y^2=0…\)

In Summary

Conic sections are geometric shapes formed by the intersection of a plane and a cone. They include circles, ellipses, parabolas, and hyperbolas. These conic sections show up in a lot of math and physics, including algebra, geometry, calculus, projectile motion, astronomy, and engineering.

Completing the square is a technique used to transform the equation of a conic section in the general form \(ax2+bxy+cy2+dx+ey+f = 0\) into the more useful standard form. The standard form provides much more useful details about the graph at a quick glance.

Topics that use Conic Sections- Completing the Square

Ellipse fitting: Conic sections, specifically ellipses, can be used to fit curves to data points. This is often used in data analysis and machine learning to fit a curve to a set of data points and make predictions about future data points.

Astronomy: Conic sections are used in celestial mechanics to describe the orbits of planets and other celestial bodies. The orbits of these bodies are typically ellipses, and conic sections can be used to accurately model and predict the movements of these objects.

Engineering: Conic sections are used in engineering to design and analyze structures and systems. For example, the shapes of car wheels and car suspension systems are often designed using conic sections.

Computer graphics: Conic sections are used in computer graphics to create and render 3D shapes and objects. For example, conic sections can be used to create and render 3D ellipsoids, which are commonly used to model objects such as planets and other celestial bodies.

Optics: Conic sections are used in optics to describe the shapes of lenses and the paths of light rays through lenses. The shapes of lenses, such as spherical and parabolic lenses, can be described using conic sections.

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