Equation And Graph Of A Circle Tangent To Y-Axis

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In this article, we will explore how to determine the equation of a circle in standard form when given its center and the information that it is tangent to the y-axis. We will also learn how to graph such a circle. This involves understanding the relationship between the circle's center, its radius, and the point of tangency. Let's dive into the problem where the center of the circle is (4, 2) and it's tangent to the y-axis.

(a) Writing the Equation of the Circle in Standard Form

To write the equation of the circle in standard form, we need two crucial pieces of information: the center of the circle and its radius. The standard form equation of a circle is given by:

(x - h)² + (y - k)² = r²

Where:

  • (h, k) represents the coordinates of the center of the circle.
  • r represents the radius of the circle.

In our case, the center of the circle is given as (4, 2). This means that h = 4 and k = 2. We now need to determine the radius, r. The problem states that the circle is tangent to the y-axis. Tangency implies that the circle touches the y-axis at exactly one point. The distance from the center of the circle to this point of tangency is the radius. Since the y-axis is a vertical line defined by x = 0, the point of tangency will have the same y-coordinate as the center of the circle. Thus, the point of tangency is (0, 2).

Now, we can calculate the radius as the distance between the center (4, 2) and the point of tangency (0, 2). We can use the distance formula, but in this case, it’s simpler to observe that the vertical distance is zero (both points have a y-coordinate of 2), and the horizontal distance is the absolute difference in the x-coordinates: |4 - 0| = 4. Therefore, the radius, r, is 4.

Now that we have the center (h = 4, k = 2) and the radius (r = 4), we can plug these values into the standard form equation of a circle:

(x - 4)² + (y - 2)² = 4²

Simplifying, we get:

(x - 4)² + (y - 2)² = 16

This is the equation of the circle in standard form. It tells us that the circle has a center at (4, 2) and a radius of 4 units. The equation clearly defines all points (x, y) that lie on the circumference of the circle. This standard form is incredibly useful because it directly reveals the circle’s key properties: its center and its radius. From this equation, we can easily visualize the circle’s position and size on a coordinate plane. The squared terms (x - 4)² and (y - 2)² indicate the shifts from the origin, and the 16 (which is 4²) gives us the squared radius, making it straightforward to identify the circle’s dimensions.

(b) Graphing the Circle

To graph the circle, we start by plotting the center, which is at the point (4, 2) on the coordinate plane. The center serves as the reference point around which the entire circle is drawn. Next, we use the radius, which we determined to be 4 units, to find several points on the circle. Since the circle is tangent to the y-axis, we know that one point on the circle is (0, 2), as this is the point of tangency. To find other points, we can move 4 units to the right, left, up, and down from the center.

  • Moving 4 units to the right from the center (4, 2) gives us the point (8, 2).
  • Moving 4 units to the left from the center (4, 2) gives us the point (0, 2) (the point of tangency).
  • Moving 4 units up from the center (4, 2) gives us the point (4, 6).
  • Moving 4 units down from the center (4, 2) gives us the point (4, -2).

These four points (8, 2), (0, 2), (4, 6), and (4, -2) lie on the circle's circumference. Plotting these points helps us visualize the circle's shape and size. Now, we can sketch the circle by connecting these points with a smooth curve. The circle should be centered at (4, 2) and extend 4 units in all directions. The accuracy of the graph depends on how smoothly we can draw the curve connecting the points.

Graphing a circle involves accurately representing its position and size on a coordinate plane. Understanding the standard form equation makes this process significantly easier. By plotting the center and using the radius to find key points, we can sketch a precise representation of the circle. The point of tangency, in this case with the y-axis, provides an additional reference to ensure the circle's placement is accurate. A well-graphed circle visually confirms the equation and provides a clear geometric representation of the algebraic expression.

In conclusion, we have successfully found the equation of the circle in standard form, which is (x - 4)² + (y - 2)² = 16, and we have learned how to graph the circle by plotting the center and using the radius to find points on the circumference. This exercise demonstrates the interplay between algebraic equations and geometric representations, crucial in understanding circles and other conic sections. The ability to derive the equation from given conditions and visualize the circle on a graph is a fundamental skill in mathematics.