Problem 62 - Algebra and Trigonometry

Question

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Circle lies in the first quadrant, tangent to both \(x\) - and \(y\) -axes; radius 5

Step-by-Step Solution

Answer

The equation is (x - 5)^2 + (y - 5)^2 = 25.

Step 1: Understanding the Problem

We need to find an equation of a circle located in the first quadrant, with a radius of 5, and tangent to both the x-axis and y-axis.

Step 2: Deciding on Circle's Center

Since the circle is tangent to both axes and in the first quadrant, the center of the circle must be exactly 5 units away from both axes. This means its center will be at the coordinates (5, 5).

Step 3: Writing the Equation of the Circle

The general equation of a circle with center (x_0, y_0) and radius r is (x - x_0)^2 + (y - y_0)^2 = r^2. Substituting (x_0, y_0) = (5, 5) and r = 5, the equation becomes (x - 5)^2 + (y - 5)^2 = 25.

Key Concepts

Radius of a Circle

The radius of a circle is a crucial concept in understanding and solving circle-related problems. The radius is the distance from the center of the circle to any point on its circumference. In this problem, we're given a radius of 5 units. This uniform distance means that every point along the circle is exactly 5 units away from the center, which is key to determining its position and ensuring it's tangent to the axes.
To recall, the formula for any circle's equation is \[(x - x_0)^2 + (y - y_0)^2 = r^2\]Where

• (x_0, y_0) are the coordinates of the center.
• r represents the radius.

Substituting the radius we have here, which is 5, gives us \[r^2 = 25\]This indicates every point on the circle is 5 units away from the center point (5, 5) as found in this problem. Understanding the radius helps us plot the circle and anticipate its interaction with the coordinate axes.

Tangent to Axes

When a circle is tangent to the axes, it means it just touches the axes without cutting across them. For tangency, the distance from the center of the circle to the axis it touches is exactly equal to the radius. In this particular exercise, the circle is tangent to both the x-axis and y-axis.
The center of this circle is (5, 5), meaning it is 5 units away from the y-axis and also 5 units away from the x-axis. Since this matches the given radius of 5, the circle doesn't overlap with the axes but merely touches them—thus being tangent. This is a perfect alignment where the radius ensures the symmetric distance precisely dictates the point of tangency.
Being tangent helps not only in defining the position of the circle relative to the axes but also provides a cue to draw accurate geometric configurations for other mathematical applications. Remember, the essence of tangency is in touching without crossing.

First Quadrant

The first quadrant of a coordinate plane is the section where both x and y values are positive. Visualizing the plane, the first quadrant lies to the right and above the origin (0, 0). In problems involving circles or other geometric shapes, specifying the quadrant helps limit where the shapes are drawn.
In our problem, the circle being located in the first quadrant implies that both the x and y coordinates of the circle's center, as well as any points on the circle, are positive integers. The determined center (5, 5) confirms this condition, ensuring all points on the circle fall within the positive range of both axes.
Thus, knowing a circle lies in the first quadrant helps logically strategize and solve the geometric puzzle it's posed in. It sets boundaries and guides the placement of shapes relative to the axes, which is crucial for accurate plotting and determining tangential properties.