Problem 61 - Algebra and Trigonometry

Question

\(55-62\) . Find an equation of the circle that satisfies the given conditions. Center \((7,-3) ; \quad\) tangent to the \(x\) -axis

Step-by-Step Solution

Answer

The equation of the circle is \((x - 7)^2 + (y + 3)^2 = 9\).

Step 1: Understand the Equation of a Circle

The standard form of the equation of a circle with a center at point \((h, k)\) and radius \(r\) is given by: \[(x - h)^2 + (y - k)^2 = r^2\]In this problem, the center is \((7, -3)\), so substituting in gives us: \[(x - 7)^2 + (y + 3)^2 = r^2\]

Step 2: Interpret the Tangency Condition

For the circle to be tangent to the \(x\)-axis, the distance from the center of the circle to the \(x\)-axis must be equal to the radius \(r\). Since the y-coordinate of the center is \(-3\), the distance from the center \((7, -3)\) to the \(x\)-axis is 3.

Step 3: Determine the Radius

Given that the circle is tangent to the \(x\)-axis, we determined in Step 2 the radius \(r\) is 3. So we substitute \(r = 3\) into the radius formula: \[(x - 7)^2 + (y + 3)^2 = 3^2\].

Step 4: Write the Final Equation

Substitute the radius back into the equation of the circle derived in Step 1, we have: \[(x - 7)^2 + (y + 3)^2 = 9\].This is the equation of the circle with center \((7, -3)\) and tangent to the \(x\)-axis.

Key Concepts

Center of a Circle

The center of a circle is a pivotal point from which all points on the circle are equidistant. This center is represented in coordinates, typically denoted as \((h, k)\) in the equation of a circle. Understanding this concept is crucial when you need to construct or analyze the equation of a circle. For instance, if the center is given as \((7, -3)\), it means that every point on the circle is a fixed distance (the radius) from this location.

• The center affects the position of the circle in the coordinate plane.
• It helps in determining other characteristics of the circle, such as its radius and area.
• The center's coordinates are used in the equation \( (x - h)^2 + (y - k)^2 = r^2 \) to help identify the circle's specific location.

Understanding that the center is key to other features helps students grasp how circles interact with geometric elements like the x-axis, as seen when the circle is tangent to an axis.

Radius of a Circle

The radius of a circle is the distance from the center of the circle to any point on its circumference. Within the context of a circle's equation, the radius squared is expressed as \(r^2\). In problems where a circle is tangent to an axis, the radius often has special significance.

• To find the radius in such cases, measure the perpendicular distance from the center to the axis of tangency.
• This distance becomes the radius if the circle is tangent to the x-axis because it just touches the x-axis and doesn’t cross it.
• For example, if the center is at \((7, -3)\), the radius is the absolute y-value, which is 3. Here, it means the circle reaches the x-axis without crossing it, making the y-distance to the axis the radius.

With the radius known, you can completely construct the circle's equation and analyze its properties.

Tangent to the x-axis

When a circle is tangent to the x-axis, it implies that the circle touches the x-axis at exactly one point. This geometric relationship influences the circle's positioning and constraints on its equation. If a circle is tangent to the x-axis, its radius can be determined directly by observing how the circle fits into the coordinate plane.

• The tangency condition provides that the vertical distance from the center (\((h, k)\)) to the x-axis equals the radius \(r\).
• For example, with a center at \((7, -3)\), the y-coordinate indicates that the circle’s radius is 3, which means the circle touches the x-axis at point \((7, 0)\).
• This property is useful in verifying and validating the equation of the circle once it's put in standard form.

Recognizing how tangency affects the radius and placement of a circle in relation to the x-axis is essential for solving and understanding these types of geometric problems.