Problem 34 - Algebra A Combined Function

Question

Hint: For Exercises 33 through 38 , first divide the equation through by the coefficient of \(x^{2}\) (or \(\left.y^{2}\right)\). $$ 2 x^{2}+2 y^{2}=18 $$

Step-by-Step Solution

Answer

The equation is a circle centered at (0, 0) with radius 3.

Step 1: Divide by the Coefficient of Quadratic Terms

The equation given is \(2x^2 + 2y^2 = 18\). Both \(x^2\) and \(y^2\) have the coefficient of 2, so divide each term in the equation by 2. This results in \(x^2 + y^2 = 9\).

Step 2: Identify the Equation Form

The equation \(x^2 + y^2 = 9\) is in the standard form of a circle equation: \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

Step 3: Determine the Radius

From the equation \(x^2 + y^2 = 9\), compare it to \(x^2 + y^2 = r^2\), so \(r^2 = 9\). Therefore, solve for \(r\) which gives \(r = \sqrt{9} = 3\).

Step 4: Conclusion

The given equation represents a circle centered at the origin \((0, 0)\) with a radius of 3.

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra and mathematics as a whole. They are polynomial equations of the second degree, typically presented in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Key features of quadratic equations:

• The highest degree of the variable is 2.
• They form a parabola when graphed on a coordinate plane.
• Solutions to quadratic equations are known as roots, which can be real or complex numbers.

Quadratic equations also play a crucial role in deriving the standard form of a circle equation. When both variables \(x\) and \(y\) in a quadratic equation are involved in forms like \(x^2 + y^2\), they can represent geometric shapes such as circles. By manipulating these quadratic terms, we can transform them into meaningful equations to identify characteristics of circles, like their centers and radii.

Radius

The radius of a circle is a crucial geometric element that defines the size of the circle. It is the distance from the center of the circle to any point on its circumference. In equations, the radius is often denoted by the letter \(r\) and plays a central role in the equation of a circle.Important points about the radius:

• The radius is always positive and greater than zero for a valid circle.
• All radii of a circle are equal in length, making them the defining measure of its size.
• In the standard form of the circle equation \(x^2 + y^2 = r^2\), the radius is \(r = \sqrt{r^2}\).

Understanding the radius helps in interpreting circle equations and identifying their sizes, both of which are important in solving geometric problems involving circles. Calculating the radius from the standard form of the equation allows one to visualize and precisely construct the circle on a coordinate plane.

Standard Form of Circle Equation

The standard form of a circle equation is instrumental in understanding and graphing circles on the coordinate plane. The typical standard form is written as \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) represents the center of the circle, and \(r\) is the radius.Here are the fundamental components:

• \((h, k)\) defines the circle's position on the coordinate plane. For example, \((0, 0)\) centers the circle at the origin.
• \(r^2\) signifies the square of the radius, determining the circle's size.
• When the equation is in the form \(x^2 + y^2 = r^2\), it reveals a circle centered at the origin.

This form provides a convenient way to identify and analyze a circle's properties quickly. Working with the standard form allows for easier manipulation and solution of various mathematical problems, especially those involving transformations of the circle such as translations or scaling on a graph.