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 circle's equation is \((x - 5)^2 + (y - 5)^2 = 25\).

Step 1: Understand the Problem

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

Step 2: Determine the Center of the Circle

A circle that is tangent to both the x-axis and the y-axis, with a radius of 5, must have its center at (5,5). This is because being tangent to the x and y axes requires the center to be equidistant from both axes, and that distance is the radius.

Step 3: Write the Equation of the Circle

The general equation of a circle with center (h, k) and radius r is:\[(x - h)^2 + (y - k)^2 = r^2\]Substitute h = 5, k = 5, and r = 5 into the formula: \[(x - 5)^2 + (y - 5)^2 = 5^2\] \[(x - 5)^2 + (y - 5)^2 = 25\]

Step 4: Finalize the Solution

We have derived the equation based on the given conditions. The circle's equation is:\[(x - 5)^2 + (y - 5)^2 = 25\]

Key Concepts

First Quadrant

The first quadrant in the Cartesian coordinate system is the region where both the x and y coordinates are positive. It is one of the four quadrants in the plane. When a circle is said to lie entirely in the first quadrant, all points on the circumference of the circle must be located where both x and y are greater than or equal to zero.

In the context of this problem, a circle that is tangent to both the x-axis and y-axis will have its center placed in such a manner that no part of the circle extends into the negative regions of the graph. This means its center must have positive x and y coordinates. Moreover, because the circle is tangent to both axes, its center will lie off the origin by a distance equivalent to its radius from both axes.

Radius

The radius of a circle is the fixed distance from its center to any point on the circle itself. The exercise states that the radius is 5, which gives us crucial information. Knowing the radius helps us determine how far the circle extends in every direction from its center.

Since the circle is tangent to both axes, if their center is (5, 5) in this specific case, each axis touches the circle exactly at a single point. Thus, the radius of 5 ensures that starting from the center (5, 5), the circle reaches the x-axis at (5, 0) and the y-axis at (0, 5). This relationship confirms the correct placement of the center at a point equidistant from both axes, making it tangent.

Tangent to Axes

A line or axis is tangent to a circle if it touches the circle at exactly one point. In this situation, the circle is tangent to both the x-axis and y-axis. This indicates that there are exactly two points of tangency: one on each axis.

For a circle to be tangent to the axes and lie in the first quadrant, a careful placement of its center is essential. The center's coordinates must be one radius length away from both axes, indicating both x and y coordinates of the circle's center are equal to the radius. Hence, for a radius of 5, the center of the circle is clearly (5, 5). This setup permits the circle to touch the x-axis at (5, 0) and the y-axis at (0, 5) without crossing into other quadrants, thereby satisfying the tangent condition.