Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(-3,6) \text {, tangent to the } y \text {-axis } $$

Step-by-Step Solution

Answer

The equation of the circle is \((x + 3)^2 + (y - 6)^2 = 9\).

Step 1: Understanding the Circle's Center and Tangency

The center of the circle is given as \( (-3, 6) \). The circle is tangent to the \( y \)-axis, meaning it touches this axis at exactly one point. The distance from the center of the circle to the \( y \)-axis is the radius of the circle.

Step 2: Determining the Radius of the Circle

The distance from the circle's center \((-3, 6)\) to the \( y \)-axis is the same as the absolute value of the \( x \)-coordinate of the center. Since the center's \( x \)-coordinate is \(-3\), the radius \( r \) is 3.

Step 3: Formulating the Circle's Equation

The standard equation for a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Substituting \( h = -3 \), \( k = 6 \), and \( r = 3 \), the equation becomes:\[(x + 3)^2 + (y - 6)^2 = 9\]

Step 4: Verifying the Condition Tangent to the y-axis

Check that the equation satisfies the tangency condition. The circle, centered at \((-3, 6)\) with a radius of 3, is exactly 3 units away from the \( y \)-axis, confirming tangency since \(|-3 - 0| = 3\), which is equal to the radius.

Key Concepts

Circle Center

The center of a circle is crucial in defining its position in the coordinate plane. It is typically denoted as a point \( (h, k) \) where \( h \) and \( k \) are the coordinates of the center. In our exercise, this point is \( (-3, 6) \). This tells us that the center of the circle is located 3 units to the left and 6 units up from the origin. \( (h, k) \) coordinates effectively anchor the circle on the graph.
When considering the center, remember:

• It indicates symmetry for the circle.
• It helps in defining the circle's equation.

If you know the center, you can easily plot the position of the circle on a graph, which is the first step to understanding how the circle interacts with other elements, like axes.

Circle Radius

The radius of a circle, represented by \( r \), is the distance from the center of the circle to any point on its edge. In this exercise, because the circle is tangent to the \( y \)-axis, the radius is the horizontal distance from the center to the \( y \)-axis.
The radius can be calculated as:

• Absolute value of the x-coordinate of the center, which is \(-3\).
• Therefore, the radius is \( | -3 | = 3 \).

This distance is consistent since tangency means touching at exactly one point, not passing through. So, understanding the radius is crucial for understanding the circle's scope and range.

Tangent to Axis

A line or axis is tangent to a circle if it touches the circle at precisely one point. In this exercise, the circle is tangent to the \( y \)-axis. This means the closest point on the circle to the \( y \)-axis is exactly at its edge, not passing through.
Key points to remember:

• The axis is considered an asymptote for the circle's expansion beyond this point.
• Any change in radius would mean either detachment from or passing through the axis.

Thus, being tangent to the \( y \)-axis pinpoints the circle's reach horizontally and clarifies its spatial interaction with the axis.

Standard Form of Circle Equation

The standard form of a circle's equation is derived from the relationship between its center and radius. This is given by the formula \( (x - h)^2 + (y - k)^2 = r^2 \).
In our example, the standard form becomes:

• Substituting \( h = -3, k = 6, \) and \( r = 3, \) leading to \( (x + 3)^2 + (y - 6)^2 = 9 \).
• This equation describes all the points (x, y) that make up the circle.

Keep in mind that this form simplifies checking the circle's properties. Once you plug in the center and the radius, the equation effectively becomes a tool to visualize or solve for any point on the circle's edge, following its symmetry around the center.