Problem 79 - Thomas Calculus

Question

Find an equation for the circle centered at \((-2,1)\) that passes through the point \((1,3) .\) Is the point \((1.1,2.8)\) inside, outside, or on the circle?

Step-by-Step Solution

Answer

The point (1.1, 2.8) is inside the circle.

Step 1: Understand the Equation of a Circle

The general equation of a circle with center \((h, k)\) and radius \(r\) is given by: \[(x - h)^2 + (y - k)^2 = r^2\]In this problem, the center of the circle is \((-2, 1)\). Substitute these values into the equation to get the partial expression: \((x + 2)^2 + (y - 1)^2 = r^2\).

Step 2: Find the Radius of the Circle

To find the radius \(r\), use the point \((1, 3)\), which lies on the circle. Substitute this into the equation found in Step 1:\[(1 + 2)^2 + (3 - 1)^2 = r^2\] Simplify to find \(r^2\): \[3^2 + 2^2 = r^2 \]\[9 + 4 = r^2\] \[r^2 = 13\]This means the radius squared is 13.

Step 3: Write the Equation of the Circle

Now that we know \(r^2 = 13\), substitute back into the equation from Step 1: \[(x + 2)^2 + (y - 1)^2 = 13\] This is the equation of the circle.

Step 4: Determine the Position of the Point (1.1, 2.8)

To find whether the point \((1.1, 2.8)\) is inside, outside, or on the circle, substitute \((x, y) = (1.1, 2.8)\) into the circle's equation:\[(1.1 + 2)^2 + (2.8 - 1)^2 = (3.1)^2 + (1.8)^2\]\[9.61 + 3.24 = 12.85\]Since 12.85 is less than 13, \((1.1, 2.8)\) is inside the circle.

Key Concepts

Center of the Circle

The center of a circle is a crucial concept when forming its equation. It is defined by a fixed point within the circle equidistant from all points on the circle's edge. In the standard circle equation, denoted by \[(x - h)^2 + (y - k)^2 = r^2,\]\(h\) and \(k\) represent the coordinates of the circle's center. The center is not just a random point; it determines the location of the entire circle in a coordinate plane.

For example, if a circle's center is at \((-2, 1)\), you insert these coordinates into the equation, resulting in \((x + 2)^2 + (y - 1)^2 = r^2\). This moves the circle to the left and up compared to a circle centered at the origin \((0, 0)\). Understanding the center helps in positioning the circle precisely where needed, which is vital in geometry challenges involving circles.

Radius of the Circle

The radius of a circle is the distance from its center to any point on its circumference. It is a constant measure and is key to determining the size of the circle.

Finding the radius involves the circle's equation. Suppose you have a circle centered at a point and a known point on the circle. You can calculate the radius \(r\) using these two points. Simply, you plug them into the circle equation to solve for \(r^2\).

• Consider a circle centered at \((-2,1)\) and passing through \((1,3)\).
• The distance between these points is computed as: \((1 + 2)^2 + (3 - 1)^2 = r^2\).
• Simplifying gives \(3^2 + 2^2 = 13\).
• The radius squared, \(r^2\), is 13, indicating the radius is the square root of 13.

Identifying the radius helps assess the circle's size, making it a fundamental aspect of geometry problems involving circles. This understanding aids in drawing or computing other properties related to the circle.

Circle Equation Format

The equation of a circle provides a structured format that outlines how a circle is expressed in algebraic terms. The standard format is \[(x - h)^2 + (y - k)^2 = r^2,\]where \((h, k)\) is the circle's center and \(r\) its radius.

This form allows us to describe and manipulate circles in the coordinate plane effortlessly. Once you've found the center and radius, you can replace \(h\), \(k\), and \(r^2\) in the template to obtain a specific equation for your circle.

• If centered at \((-2, 1)\) with a radius squared of 13, the equation is \((x + 2)^2 + (y - 1)^2 = 13\).
• This pattern makes it easy to identify the circle's size and position.
• Such equations help determine other points' positions relative to the circle, checking if they lie inside, outside, or on the circle.

Understanding and applying this format is essential in many geometric contexts, as it provides a simple yet powerful way to work with circles on a coordinate plane.