Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 35-46: Find an equation of the circle that satisfies the stated conditions. $$ \text { Center } C(-4,6), \text { passing through } P(1,2) $$

Step-by-Step Solution

Answer

The equation of the circle is \((x + 4)^2 + (y - 6)^2 = 41\).

Step 1: Understand the Standard Form of a Circle's Equation

The standard form for the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Identify the Center of the Circle

The center of the circle is given as \(C(-4, 6)\). This means that \(h = -4\) and \(k = 6\).

Step 3: Use the Point to Find the Radius

The circle passes through the point \(P(1, 2)\), which allows us to calculate the radius by using the distance formula between the center \(C\) and the point \(P\).

Step 4: Calculate the Radius

The distance formula for radius \(r\) is: \[r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]. Substituting the center \((-4, 6)\) and point \((1, 2)\), we find the radius: \[r = \sqrt{(1 - (-4))^2 + (2 - 6)^2} = \sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41}\].

Step 5: Form the Equation of the Circle

Substitute \(h = -4\), \(k = 6\), and \(r^2 = 41\) into the standard form equation. The equation becomes: \[(x + 4)^2 + (y - 6)^2 = 41\].

Key Concepts

Center of the Circle

To find the equation of a circle, knowing its center is crucial. A circle's center is the point inside it that is exactly equidistant from all points along the circle's circumference. In the standard form of a circle’s equation \((x - h)^2 + (y - k)^2 = r^2\),

• \(h\) represents the x-coordinate of the center.
• \(k\) is the y-coordinate.

Finding the center of a circle within a problem involves identifying these coordinates. For instance, in the problem at hand, the center is given as \(C(-4, 6)\). This tells us that our center is positioned 4 units left and 6 units up from the origin point in a standard Cartesian coordinate system. Understanding this center point helps in both calculating the circle’s radius and forming its equation.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the circle. Knowing the radius helps in forming the equation of a circle. To find the radius, you often need another point on the circle, besides the center.In our example, the circle passes through point \((1, 2)\), and the center is \((-4, 6)\). You use the distance formula to calculate the radius. The calculation involves:

• Subtracting the x-coordinates and squaring the result: \(1 - (-4) = 5\).
• Subtracting the y-coordinates and squaring the result: \(2 - 6 = -4\).
• Adding these squares: \(5^2 + (-4)^2 = 25 + 16 = 41\).
• Taking the square root of the sum to find the radius: \(r = \sqrt{41}\).

Thus, the radius of the circle is \(\sqrt{41}\), providing one of the essential parts of the circle's equation.

Standard Form of a Circle's Equation

A circle can be uniquely described by its standard form equation \((x - h)^2 + (y - k)^2 = r^2\). This form encapsulates:

• \((h, k)\), the center coordinates.
• \(r\), the radius, crucially squared in the equation.

To plug in our particular values from the problem:

• The center \((-4, 6)\) gives us \(h = -4\) and \(k = 6\), which transforms our equation to \((x + 4)^2 + (y - 6)^2 = r^2\).
• With a radius squared \(r^2 = 41\), we end up with the equation: \[(x + 4)^2 + (y - 6)^2 = 41\].

This equation can be used to identify any point that lies on the circle, simply by replacing the values of \(x\) and \(y\) and checking if the equation holds true. Understanding this form allows for versatile applications in geometry and problem-solving.

Distance Formula

The distance formula helps calculate the length of a line segment between any two points \((x_1, y_1)\) and \((x_2, y_2)\) on the Cartesian plane. It is expressed as:\[r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is deeply rooted in the Pythagorean theorem, enabling us to find the distance between any two points or, as in this scenario, calculate the radius of a circle.By substituting point \((1, 2)\) for \((x_2, y_2)\) and center \((-4, 6)\) for \((x_1, y_1)\), we compute:

• The change in x: \(x_2 - x_1 = 1 - (-4) = 5\).
• The change in y: \(y_2 - y_1 = 2 - 6 = -4\).
• Calculate the distance: \(\sqrt{5^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41}\).

Thus, the line's length—or our circle's radius—is \(\sqrt{41}\). Understanding the distance formula is vital for calculating or verifying values in geometric contexts.