Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. Tangent to both axes, center in the second quadrant, radius 4

Step-by-Step Solution

Answer

The equation of the circle is (x + 4)^2 + (y - 4)^2 = 16.

Step 1: Understand the Problem

We need to find the equation of a circle that is tangent to both the x-axis and y-axis, with its center in the second quadrant and a given radius of 4. A circle is tangent to an axis if the distance between the circle's center and the axis is equal to the radius.

Step 2: Identify the Center of the Circle

Since the circle is tangent to the x-axis and the y-axis and its center is in the second quadrant, the x-coordinate and y-coordinate of the center must both be equal to the radius but must adhere to the quadrant rules. Thus, the coordinates are (-r, r). Given r = 4, the center is (-4, 4).

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. Substituting h = -4, k = 4, and r = 4, the equation becomes (x + 4)^2 + (y - 4)^2 = 16.

Step 4: Simplify the Equation

Expand the equation to verify correctness. Starting with (x + 4)^2 + (y - 4)^2 = 16. Both terms are expanding correctly into the equation of a circle’s general form.

Key Concepts

Tangent to Axes

A circle is tangent to an axis if it just touches that axis without crossing it. In simpler terms, it lies just alongside the axis, making a single point of contact.
Since a circle is perfectly round, the entire shape is equidistant from its center to any point on its surface. This unique property helps us determine the position of the circle as it relates to the axes.
The circle's distance from the axis it is tangent to is exactly equal to the radius. Therefore, if we know the radius, we can find the distance from the center to any axis it touches. This becomes particularly useful in determining the circle's precise positioning on the plane.

Quadrants

The Cartesian coordinate plane is divided into four quadrants, divided by the x-axis and y-axis. Each quadrant has a distinct combination of positive and negative values for x and y.
The second quadrant, where the circle's center is supposed to be, has negative x-values and positive y-values. To visualize:

• First Quadrant: (+x, +y)
• Second Quadrant: (-x, +y)
• Third Quadrant: (-x, -y)
• Fourth Quadrant: (+x, -y)

The center of our circle needs to reflect these sign conventions. Hence, while calculating the coordinates of the center, they should display a negative x-coordinate and a positive y-coordinate.

Equation Derivation

The equation of a circle is derived using its center and radius. The standard form of a circle's equation is \[(x - h)^2 + (y - k)^2 = r^2\] where

• (h, k) is the circle's center
• r is the radius

For our circle with center (-4, 4) and radius 4, we substitute these values into the formula.
This gives us: \[(x + 4)^2 + (y - 4)^2 = 16\].
Each term inside the parentheses reflects opposite signs to the center coordinates due to the subtraction in the formula. The resulting equation perfectly represents our circle given all constraints.

Radius of a Circle

The radius is the distance from the center of the circle to any point on its boundary. In the context of this problem, it ensures the circle's tangency to both axes.
Since the radius, r, is provided as 4, this indicates two things:

• The center should be 4 units away from both the x-axis and y-axis.
• It maintains the tangent property where the center of the circle, at (-4, 4), ensures that the circle just touches both axes.

The fixed radius directly impacts the positioning and size of the circle, crucial for mathematical computation as it guides the derivation of the equation.