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Chapter 10: Problem 40

Identify the conic section and use technology to graph it. $$x^{2}+y^{2}-4 x+2 y-7=0$$

Short Answer

Expert verified
Given equation: $$x^2 + y^2 - 4x + 2y - 7 = 0$$ Answer: The given equation represents a circle with its center at (2, -1) and radius of sqrt(12).

Step by step solution

01

Rewrite the equation in its standard form

To rewrite the equation in a standard form for quadratic equation, complete the squares for the x and y terms. $$x^{2}+y^{2}-4x+2y-7=0$$ Rearrange the equation by grouping the x and y terms. $$ (x^2 - 4x) + (y^2 + 2y) = 7$$ Now, complete the squares for the x and y terms. $$ (x^2 - 4x + 4) + (y^2 + 2y + 1) = 7 + 4 + 1$$ Now rewrite it as : $$ (x - 2)^2 + (y + 1)^2 = 12$$
02

Identify the conic section type

Observe that the equation is of the form: $$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$ where h = 2, k = -1, a^2 = 12 and b^2 = 12. Since a^2 = b^2, it means a = b. This is the equation of a circle with center (h, k) and radius r = a = b.
03

Use technology to graph the conic section

To graph the conic section, use a graphing tool like Desmos, Geogebra, or a graphing calculator. Input the equation in the standard form: $$(x - 2)^2 + (y + 1)^2 = 12$$ This will plot the graph of the circle with center at (2, -1) and radius of sqrt(12).

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Key Concepts

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

Standard Form of a Quadratic Equation
Understanding the standard form of a quadratic equation is crucial for graphing conic sections. A quadratic equation in two variables, x and y, typically represents different types of conic sections. The general standard form is written as
\(Ax^2 + By^2 + Cx + Dy + E = 0\),
where A, B, C, D, and E are constants. Depending on the values of these constants, the equation represents different conic sections - parabolas, circles, ellipses, or hyperbolas. For circles, both coefficients of the squared terms, A and B, are equal and non-zero, without any xy term present. Recognizing this pattern is key to identifying the conic section before graphing it.
Completing the Square
When graphing conic sections, 'completing the square' is a method used to transform equations into their standard form. The technique involves adding and subtracting certain values to create perfect square trinomials from the x and y terms.
For example, to complete the square for the x-terms in a quadratic equation, you would:
  • Group the x-terms together and factor out the coefficient if necessary.
  • Take half the coefficient of the x-term, square it, and add it inside the parenthesis.
  • Balance the equation by adding the same value to the other side.

By doing this for both x and y terms, you can rewrite the equation so it clearly reveals the conic section's type and characteristics, such as the center and radius of a circle.
Circle Equations
Circle equations in their standard form provide an explicit description of a circle's size and position in a coordinate plane. The standard form for the equation of a circle with center at point \( (h, k) \) and radius r is:
\( (x - h)^2 + (y - k)^2 = r^2 \).
Here, \( (h, k) \) represents the center of the circle, and r is the radius. This form is derived using 'completing the square' on the general quadratic equation and is highly useful because it allows for easy identification of the circle's properties just by inspection of the equation.
Conic Sections Identification
Identifying conic sections from their equations is a fundamental skill in algebra and geometry. Examining the coefficients and the structure of a quadratic equation helps to determine the type of conic section it represents. A circle has equal coefficients for \(x^2\) and \(y^2\) without an xy term, an ellipse has different coefficients, a parabola has a missing squared term, and a hyperbola has opposite-signed squared terms. Recognizing these patterns not only allows for proper identification but also informs the specific method for graphing the conic section on a coordinate plane.

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Most popular questions from this chapter

(a) Graph the curve given by \(x=\sin k t \quad\) and \(\quad y=\cos t \quad(0 \leq t \leq 2 \pi)\) when \(k=1,2,3,\) and \(4 .\) Use the window with \(-1.5 \leq x \leq 1.5 \quad\) and \(\quad-1.5 \leq y \leq 1.5\) and \(t\) -step \(=\pi / 30\) (b) Without graphing, predict the shape of the graph when \(k=5\) and \(k=6 .\) Then verify your predictions graphically.

(a) Graph these hyperbolas (on the same screen if possible \()\) $$\frac{y^{2}}{4}-\frac{x^{2}}{1}=1 \quad \frac{y^{2}}{4}-\frac{x^{2}}{12}=1 \quad \frac{y^{2}}{4}-\frac{x^{2}}{96}=1$$ (b) Compute the eccentricity of each hyperbola in part (a). (c) On the basis of parts (a) and (b), how is the shape of a hyperbola related to its eccentricity?

Use a calculator in degree mode and assume that air resistance is negligible. A golf ball is hit off the ground at an angle of \(\theta\) degrees with an initial velocity of 100 feet per second. (a) Graph the path of the ball when \(\theta=20^{\circ}, \theta=40^{\circ}\) \(\theta=60^{\circ},\) and \(\theta=80^{\circ}\) (b) For what angle in part (a) does the ball land farthest from where it started? (c) Experiment with different angles, as in parts (a) and (b), and make a conjecture as to which angle results in the ball landing farthest from its starting point.

Sketch the graph of the equation. $$r=1 / \theta \quad(\theta > 0)$$

Use the information given in Special Topics 10.3. A and summarized in the endpapers at the beginning of this book to find a parameterization of the conic section whose rectangular equation is given. Confirm your answer by graphing. $$\frac{(x-2)^{2}}{10}+\frac{(y+5)^{2}}{20}=1$$
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