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The standard form of an ellipse equation is \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]where:

• \((h, k)\) is the center of the ellipse,
• \(a\) is the semi-major axis length,
• \(b\) is the semi-minor axis length.

When the right-hand side of the equation equals 1, and both terms are fractions, it indicates the equation is in its standard form.This form is useful because it immediately tells us the ellipse's center and axes lengths. The major axis is the longer one, stretching across 2\(a\), while the minor axis stretches across 2\(b\). Using this information, sketching an ellipse becomes straightforward. We'll identify equation parts and transform them to this form in the next sections.

Completing the square is a mathematical process used to transform a quadratic expression into a perfect square trinomial.This step is crucial in rewriting the given ellipse equation into its standard form.In the example given, we started with:\[x^2 + 4y^2 + 4x + 32y + 64 = 0\]First, group the terms with the same variables:

• \((x^2 + 4x)\)
• \((4y^2 + 32y)\)

Next, for each group, add and subtract numbers to complete the square.For \(x^2 + 4x\): add \((4 = (4/2)^2)\) to form \((x+2)^2\).For \(4(y^2 + 8y)\): factor out the 4, then complete the square for \(y\) by adding \((16 = (8/2)^2)\) to get \(4(y+4)^2\).Finally, this gives:\[(x+2)^2 + 4(y+4)^2 = 4\]The result after completing the square is an expression that can be easily converted to the standard form.

Graphing an ellipse from its standard form can be straightforward.First, determine the center of the ellipse from the equation.In our case, the equation was transformed into:\[\frac{(x+2)^2}{4} + (y+4)^2 = 1\]From this, the center is at \((-2, -4)\).Identify the lengths of the semi-major and semi-minor axes:

• Semi-major axis \(a = 2\)
• Semi-minor axis \(b = 1\)

To graph:

• Start at the center.
• Move \(a\) units along the x-axis both left and right.
• Move \(b\) units along the y-axis both up and down.

Connect these points in an oval shape and there you have an ellipse.This process makes graphing much simpler than guessing or checking.

Transforming an equation into the standard form of an ellipse is a critical step.The initial equation is often not in a form that's easy to interpret.For example, we started from:\[x^2 + 4y^2 + 4x + 32y + 64 = 0\]The transformation starts by grouping terms to focus on each variable separately.Then, through completing the square, we reformed the separate group equations into perfect squares.Such as transforming \(x^2 + 4x\) into \((x+2)^2\).Once added to the equivalent expression for \(y\) and factoring out coefficients, you’re ready to convert to standard form by scaling.Finally, place it into the form:\[\frac{(x+2)^2}{4} + \frac{(y+4)^2}{1} = 1\]Understanding these steps helps solidify your grasp on ellipse equations.Mastering this lets you switch between forms and interpret characteristics quickly.