Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Tangent to both axes, center in the fourth quadrant, radius } 3 $$

Step-by-Step Solution

Answer

The equation of the circle is \\( (x-3)^2 + (y+3)^2 = 9 \\).

Step 1: Identify the conditions for the circle

The problem states that the circle is tangent to both the x-axis and y-axis, its center is in the fourth quadrant, and it has a radius of 3. We'll use these conditions to set up the coordinates for the center of the circle.

Step 2: Determine the coordinates of the center

Since the circle is tangent to the x-axis and y-axis, the distance from the center to each axis must equal the radius, which is 3. The center of the circle is thus 3 units away from both the x and y axes.

Step 3: Choose the correct quadrant for the center

The fourth quadrant means the x-coordinate will be positive and the y-coordinate will be negative. Therefore, the center of the circle, given it is tangent to both axes, is (3, -3).

Step 4: Write the equation of the circle

The equation for a circle with center \(h, k\) and radius \r\ is \( (x-h)^2 + (y-k)^2 = r^2\). Given \(h = 3, k = -3\), and \(r = 3\), the equation becomes \((x-3)^2 + (y+3)^2 = 9\).

Key Concepts

Tangent to Axes

A circle is said to be tangent to an axis if it touches the axis at exactly one point. Imagine the circle just kissing the x-axis or y-axis, where the edge of the circle just barely makes contact. When a circle is tangent to both the x-axis and the y-axis, like in this problem, the center of the circle is positioned such that the distance to each axis equals the radius. This implies an equal distance to both axes.

• For a circle tangent to the x-axis, the y-coordinate of the circle's center is exactly the radius.
• For a circle tangent to the y-axis, the x-coordinate equals the radius.

When these conditions are satisfied, it ensures the circle is perfectly balanced between both axes, not crossing them, but just touching them. In essence, this placement creates a "tangent point" with each axis.

Circle Center Coordinates

The coordinates of the center of a circle are crucial, as they determine where the circle is placed within a two-dimensional plane. For a circle tangent to both axes, like in this exercise, the coordinates are derived from its tangency condition as well as its radius.

• The x-coordinate of the center is equal to the radius when tangent to the y-axis.
• The y-coordinate of the center equals the radius when tangent to the x-axis.

In this problem, the circle's center lies in the fourth quadrant. This quadrant necessitates that the x-coordinate is positive and the y-coordinate is negative, therefore making the center's coordinates \((3,-3)\). Knowing this, we place our circle accurately in its intended spot.

Radius

The radius of a circle is the distance from its center to any point on the circle itself. In this particular problem, the radius is given as 3. This measure is a constant distance that determines not only the size of the circle, but also its placement when tangent to the axes. When a circle has a specific radius and is tangent to the axes, this distance rules how far the center is placed from each axis—exactly 3 units in this case:

• The center is 3 units away from the x-axis.
• It is also 3 units away from the y-axis.

This ensures the circle is perfectly snug, touching both axes appropriately without crossing them. The uniform radius implies that all points on the circle remain precisely 3 units from the center no matter the direction.

Quadrants in Graphing

The Cartesian plane is divided into four sections known as quadrants, which help in identifying the positions of points. These quadrants are numbered counterclockwise starting from the upper-right section, known as the first quadrant. For this problem, we focus on the fourth quadrant.In the fourth quadrant:

• The x-coordinates of points are positive.
• The y-coordinates are negative.

This unique sign arrangement allows for the graph to represent different shapes and patterns when dealing with points on the circle. With the center of the circle located at \((3, -3)\), it falls perfectly within this fourth quadrant, adhering to the characteristic signs that define point positioning in this space.