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Chapter 10: Problem 39

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. $$4 x^{2}-16 x+9 y^{2}+36 y=-16$$

Short Answer

Expert verified
The equation represents an ellipse.

Step by step solution

01

Rewrite the Equation

Start by rewriting the given equation in a more standard form. The given equation is \(4x^2 - 16x + 9y^2 + 36y = -16\). First, we need to move the constant term to the right: \[4x^2 - 16x + 9y^2 + 36y + 16 = 0.\]
02

Complete the Square for x-terms

Focus on the \(x\)-terms: \(4x^2 - 16x\). Factor out the 4: \[4(x^2 - 4x).\] To complete the square, take half of the coefficient of \(x\), which is \(-4\), giving \(-2\). Square it to get 4, and write \((x-2)^2\) inside the parentheses: \[4((x-2)^2 - 4).\] Thus, this becomes \[4(x-2)^2 - 16.\]
03

Complete the Square for y-terms

Proceed with the \(y\)-terms: \(9y^2 + 36y\). Factor out the 9: \[9(y^2 + 4y).\] Take half of the coefficient of \(y\), which is 4, giving 2. Square it to get 4 and write \((y+2)^2\) inside the parentheses: \[9((y+2)^2 - 4).\] Thus, this becomes \[9(y+2)^2 - 36.\]
04

Combine and Simplify the Equation

Substitute back the completed squares into the equation: \[4(x-2)^2 - 16 + 9(y+2)^2 - 36 + 16 = 0.\] Simplify the equation: \[4(x-2)^2 + 9(y+2)^2 = 36.\]
05

Identify the Type of Conic

Now, focus on identifying the type of conic section. The equation \[ \frac{(x-2)^2}{9} + \frac{(y+2)^2}{4} = 1 \] is in the standard form of an ellipse (\(Ax^2 + By^2 = 1\) with positive coefficients for both variables). This confirms that the original equation represents an ellipse.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabolas
A parabola is a symmetrical, open plane curve formed by the intersection of a cone with a plane parallel to its side. It can also be understood as the graph of a quadratic function, typically in the form of \[ y = ax^2 + bx + c \] Characteristics of parabolas include:
  • They have a vertex, which is the highest or lowest point on the graph, depending on orientation.
  • The axis of symmetry passes through the vertex, dividing the parabola into two mirror-image halves.
  • They can open upwards, downwards, or sideways (left or right).
In contrast to other conic sections, parabolas are defined by having only one squared term. This makes them relatively easy to recognize. Remember that parabolas opening up or down have a vertical axis, whereas those opening sideways have a horizontal one.
Circles
A circle, described by points equidistant from a center point, is a special type of an ellipse. Its general equation is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center and \(r\) is the radius.Key properties include:
  • All points on the circle are the same distance from the center.
  • The distance, known as the radius, remains constant.
  • The circle is symmetric about its center point.
In terms of graph identification, the presence of equal coefficients in front of the squared terms \(x^2\) and \(y^2\) indicates a circle. That's unlike an ellipse, which usually has differing coefficients. It's simple, yet unique structure makes it stand out among conic sections.
Ellipses
Ellipses are defined as squashed or stretched circles. They have the equation form:\[ \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \] where \((h, k)\) is the center, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.Characteristics of an ellipse include:
  • Two axes of symmetry: a major axis and a minor axis.
  • Foci inside the ellipse where the sum of the distances from all points on the ellipse to each focus remains constant.
  • Ellipses can look like circles when \(a=b\), but generally, the coefficients differ, providing the stretch.
In the given exercise, we concluded an ellipse based on the equation's form after completing the square. Recognizing ellipses is all about identifying this standard form.
Hyperbolas
Hyperbolas are conic sections formed by the intersection of a double cone with a plane in a manner that is neither parallel nor perpendicular to the cone's axis. Their standard form is:\[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \]or\[ \frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1 \]where \((h, k)\) marks the center, and the \(a\) and \(b\) values help shape the hyperbola.Key features of hyperbolas are:
  • Two branches, unlike other conics, each curving away from the center.
  • Asymptotes, which are lines that the hyperbola approaches but never touches.
  • Vertically or horizontally oriented, depending on the placement of terms.
Hyperbolas can be tricky because they have two squared terms like ellipses, but differ in having opposite signs, making them distinct in contrast to the other conics.

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