Problem 13 - Precalculus with Limits

Question

In Exercises 7-14, determine whether each point lies on the graph of the equation. \( x^2 + y^2 = 20 \) (a) \( (3, -2) \) (b) \( (-4, 2) \)

Step-by-Step Solution

Answer

The point (3, -2) does not lie on the graph of the equation, but the point (-4, 2) does.

Step 1: Substitute (3, -2) in the equation

Replace x with 3 and y with -2 in the equation \( x^2 + y^2 = 20 \). After making the substitutions, the equation becomes \( 3^2 + (-2)^2 \). Computing each term gives us \( 9 + 4 \), which equals 13.

Step 2: Check the result for (3, -2)

The result, 13, does not equal the right-hand side of the equation, 20. Thus, (3, -2) does not lie on the graph of the equation.

Step 3: Substitute (-4, 2) in the equation

Replace x with -4 and y with 2 in the equation \( x^2 + y^2 = 20 \). After making the substitutions, the equation becomes \( (-4)^2 + 2^2 \). Computing each term gives us \( 16 + 4 \), which equals 20.

Step 4: Check the result for (-4, 2)

The result, 20, equals the right-hand side of the equation, 20. Thus, (-4, 2) does lie on the graph of the equation.

Key Concepts

Circle Equation

Understanding the equation of a circle is pivotal in graphing and analyzing circles in coordinate geometry. The general form of a circle equation is \( x^2 + y^2 = r^2 \) where \( (x, y) \) are the coordinates of a point on the circle and \( r \) is the radius. The center of this circle is at the origin point \( (0, 0) \) of the coordinate system.

For instance, the given equation in the exercise \( x^2 + y^2 = 20 \) represents a circle centered at the origin with a radius squared of 20. To find the actual radius length, you would take the square root of 20. The beauty of circles in this form is their symmetry; they have uniform distance from the center to any point on their edge.

Coordinate Points

Coordinate points are the bread and butter of graphing on the Cartesian plane. They provide a way to locate and represent an exact position as a pair of numbers, \( (x, y) \). In this system, \( x \) represents the horizontal position, and \( y \) represents the vertical position.

Plotting Points

Plotting points like \( (3, -2) \) or \( (-4, 2) \) involves moving right or left from the origin for the \( x \) value, and up or down for the \( y \) value. It’s crucial to get the order correct: it’s always \( (x, y) \) and not \( (y, x) \). By plotting points on the graph, you can visualize geometric shapes and solve problems involving distance, midpoint, slope, and more.

Substitution Method

The substitution method is a foundation technique in algebra used for solving systems of equations, or checking whether points lie on a graph, as in this exercise. It's about replacing variables with their respective values to simplify an equation or compare results.

Applying Substitution

When substituting the coordinate points \( (3, -2) \) and \( (-4, 2) \) into the circle equation, you replace the \( x \) and \( y \) variables with the corresponding numbers from each point. This allows you to determine if the points satisfy the equation, or in a visual sense, if they 'lie' on the circle. Correct substitution requires careful attention to positive and negative signs, and consistently following the order of operations to evaluate the result accurately.