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Chapter 2: Problem 61

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

Short Answer

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

Step by step solution

01

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: \[(x - 7)^2 + (y + 3)^2 = r^2\]
02

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.
03

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: \[(x - 7)^2 + (y + 3)^2 = 3^2\].
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檛 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鈥檚 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.

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Most popular questions from this chapter

Manufacturing Cost The manager of a furniture factory finds that it costs \(\$ 2200\) to manufacture 100 chairs in one day and \(\$ 4800\) to produce 300 chairs in one day. (a) Assuming that the relationship between cost and the number of chairs produced is linear, find an equation that expresses this relationship. Then graph the equation. (b) What is the slope of the line in part (a), and what does it represent? (c) What is the \(y\) -intercept of this line, and what does it represent?

Find an equation of the line that satisfies the given conditions. Slope \(3 ; \quad y\) intercept \(-2\)

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cost of Driving The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May her driving cost was \(\$ 380\) for 480 \(\mathrm{mi}\) and in June her cost was \(\$ 460\) for 800 \(\mathrm{mi}\) . Assume that there is a linear relationship between the monthly cost \(C\) of driving a car and the distance driven \(d\) (a) Find a linear equation that relates \(C\) and \(d\) (b) Use part (a) to predict the cost of driving 1500 \(\mathrm{mi}\) per month. (c) Draw the graph of the linear equation. What does the slope of the line represent? (d) What does the \(y\) -intercept of the graph represent? (d) Why is a linear relationship a suitable model for this situation?
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