Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determining Solution Points In Exercises 59 and 60 determine which point lies on the graph of the circle. (There may be more than one correct answer.) \((x+2)^{2}+(y-3)^{2}=25\) (a) (-2,3) (b) (0,0) (c) (1,-1) (d) \((-1,3-2 \sqrt{6})\)

Step-by-Step Solution

Answer

The points that lie on the graph of the circle are (1, -1) and (-1, 3 - 2 \sqrt{6}).

Step 1: Substituting Point (a)

Substitute the coordinates of the first point, (-2, 3), into the equation: \((-2+2)^{2}+(3-3)^{2}=0^{2}+0^{2}=0\). Because this result doesn't equal to the right side of the equation, the point (-2, 3) does not lie on the graph of the circle.

Step 2: Substituting Point (b)

For the second point (0,0), we plug in the coordinates in the equation: \((0+2)^{2}+0-3^{2}=4+(-3)^{2}=4+9=13\). Because this result is not equal to 25, the point (0,0) does not lie on the circle's graph either.

Step 3: Substituting Point (c)

Substitute the coordinates of point (1, -1) into the equation: \((1+2)^{2}+(-1-3)^{2}=3^{2}+(-4)^{2}=9+16=25\). As this equals 25, the point (1, -1) does lie on the circle's graph.

Step 4: Substituting Point (d)

Substitute the coordinates of the point (-1, 3-2 \sqrt{6}) into the equation: \( (-1+2)^{2}+(3-2\sqrt{6}-3)^{2}=1^{2}+(-2\sqrt{6})^{2}=1+24=25\). Therefore, the point (-1, 3 - 2\sqrt{6}) also lies on the circle's graph.

Key Concepts

Coordinate Substitution

To determine if a given point lies on the graph of a circle, we use the method of coordinate substitution. This involves plugging the coordinates of the point into the circle's equation. If the resulting expression is true, then the point lies on the circle.
To illustrate, let's consider a circle with the equation \((x+2)^{2}+(y-3)^{2}=25\). Each point given in the exercise has its coordinates substituted into this equation to see if it results in a true statement.
The process is simple:

• Replace "x" and "y" in the equation with the coordinates of the point.
• Simplify the equation to see if both sides are equal.

If substituting a point results in the left-hand side of the equation being equal to the right-hand side (in this case 25), then the point lies on the circle. Coordinate substitution is therefore a vital tool in checking the position of points relative to a circle.

Graph of a Circle

The graph of a circle is the set of all points in a plane that are at a fixed distance from a center point. This distance is known as the radius.
The visual representation of a circle consists of its center, radius, and circumference. Each point along the circumference satisfies the circle's equation, indicating it is equidistant from the center.
For the equation \(x+2)^{2}+(y-3)^{2}=25\), the center of the circle is at (-2, 3) and the radius is 5 (since \( \sqrt{25} = 5\)).
Important characteristics of the graph include:

• The circle's center, which is shifted from the origin based on the equation \( (x-h)^{2}+(y-k)^{2}=r^{2}\).
• The radius, determining the size of the circle.
• The boundary, which includes all points satisfying the equation.

Understanding the graph of a circle helps in visualizing and solving geometrical problems related to distance and location on a plane.

Equation of a Circle

The equation of a circle in a standard form is written as \( (x-h)^{2}+(y-k)^{2}=r^{2}\), where (h, k) is the center of the circle and r is the radius.
This format helps in immediately identifying critical attributes of the circle:

• Center: The coordinates (h, k) indicate where the circle’s midpoint is located on the plane.
• Radius: The value of r is the consistent distance from the center to any point on the circumference.
• Symmetry: The equation reflects the perfect symmetry of a circle relative to its center.

To find if a point is on the circle, substitute the x and y values into the equation. If it holds true, the point is on the circle.
Understanding this equation is crucial for graphing circles and solving more complex problems that involve intersections, tangents, and chords. This knowledge provides the foundational understanding needed to progress to more advanced geometric concepts.