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Chapter 10: Problem 66

Classifying a Conic from a General Equation, classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$ 9 x^{2}+4 y^{2}-90 x+8 y+228=0 $$

Short Answer

Expert verified
The graph of the given equation is an ellipse.

Step by step solution

01

Grouping terms

Group the x-terms and the y-terms together to form two squared binomials. The given equation is \(9x^{2}-90x+4y^{2}+8y+228=0\). Rearranging, this becomes \(9(x^{2}-10x)+4(y^{2}+2y)=-228\)
02

Completing the Square

Here we need to complete the square for both the x-terms and the y-terms in order to get the equation in standard form. For the x-terms, we take half of -10 (the coefficient of x) and square it to get 25. We add and subtract this inside the parenthesis. Similarly, for the y-terms, we take half of 2 (coefficient of y) and square it to get 1. We add and subtract this inside the parenthesis as well. So the equation becomes \(9[(x-5)^{2}-25]+4[(y+1)^{2}-1]=-228\)
03

Reordering and Simplifying

Now, move the constants outside the brackets and combine like terms. The equation becomes \(9(x-5)^{2}-225+4(y+1)^{2}-4=-228\), which then simplifies to \(9(x-5)^{2}+4(y+1)^{2}=1\)
04

Determine the Type of Conic

In the resulting equation, both x- and y-terms are squared, the coefficients of the x- and y-terms are both positive, but they are not equal. So this is an equation of an ellipse.

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Key Concepts

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

Ellipse
An ellipse is a type of conic section that looks like a stretched circle. It has two main axes: the major axis, which is the longest, and the minor axis, which is shortest. The points on an ellipse are equidistant from two fixed points known as the foci.
  • The equation of an ellipse is generally written in the form: \[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]
  • Here, \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
To determine if a conic section is an ellipse, both the x and y terms should be squared and have positive coefficients. In the case when the coefficients are equal, it's a circle
understanding the properties of an ellipse can help in visualizing its shape and orientation.
Completing the Square
Completing the square is a mathematical technique used to transform a quadratic expression into a perfect square trinomial. This method is very useful in solving quadratic equations and in the process of conics classification.
To complete the square for an expression like \(ax^2+bx\):
  • Take half of the coefficient of \(x\), \(\frac{b}{2}\), square it.
  • Add and subtract this square inside the expression to keep it balanced.
For example, with \(x^2 - 10x\), take half of \(-10\) to get \(-5\). Squaring it gives \(25\). Add and subtract \(25\) to complete the square: \((x-5)^2 - 25\).
This technique is handy as it helps in rearranging the conic equation into its standard form.
Classifying Conics
Conic sections are curves obtained by intersecting a plane with a double-napped cone. Depending on the angle of the intersecting plane, various conic shapes can be formed: such as circles, ellipses, parabolas, and hyperbolas.
To classify a conic section from its general equation, analyze the squared terms:
  • If both x and y are squared, and their coefficients are positive and unequal, it's an ellipse.
  • If both x and y are squared with equal coefficients, it's a circle.
  • If only one variable is squared, it's a parabola.
  • If both x and y are squared but with opposite signs, it's a hyperbola.
Classification is important as it dictates the methods used for further calculations and the possible applications in real-world scenarios.

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Most popular questions from this chapter

Converting a Polar Equation to Rectangular Form In Exercises \(91-116,\) convert the polar equation to rectangular form. $$r^{2}=\cos 2 \theta$$

Converting a Polar Equation to Rectangular Form In Exercises \(91-116,\) convert the polar equation to rectangular form. $$\theta=11 \pi / 6$$

Converting a Polar Equation to Rectangular Form Convert the polar equation $$r=2(h \cos \theta+k \sin \theta)$$ to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.

Converting a Polar Equation to Rectangular Form In Exercises \(91-116,\) convert the polar equation to rectangular form. $$r=-3 \sec \theta$$

In Exercises \(23-48\) , sketch the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $$r=-3 \pi / 4$$
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