Problem 40 - Contemporary Precalculus

Question

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

Step-by-Step Solution

Answer

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 1: 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 : $$ (x - 2)^2 + (y + 1)^2 = 12$$

Step 2: 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.

Step 3: 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).

Key Concepts

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.