Problem 41 - Precalculus

Question

Specify the center and radius of each circle. Also, determine whether the given point lies on the circle. $$(x+8)^{2}+(y-5)^{2}=13 ;(-5,2)$$

Step-by-Step Solution

Answer

Center: (-8, 5); Radius: \(\sqrt{13}\); Point (-5, 2) is not on the circle.

Step 1: Identify the center and radius of the circle

The equation of a circle in standard form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Comparing \((x+8)^2 + (y-5)^2 = 13\) to the standard form, we can identify that the center \((h, k)\) is \((-8, 5)\) and \(r^2\) is 13. So, the radius \(r\) is \(\sqrt{13}\).

Step 2: Determine if the point lies on the circle

To check if the point \((-5, 2)\) lies on the circle, substitute \(x = -5\) and \(y = 2\) into the circle's equation \((x+8)^2 + (y-5)^2 = 13\). Calculate:- \((-5 + 8)^2 + (2 - 5)^2 = (3)^2 + (-3)^2 = 9 + 9 = 18.\)Since the left-hand side equals 18 and not 13, \((-5, 2)\) does not lie on the circle.

Key Concepts

Standard Form of a Circle

The standard form of the equation of a circle is a fundamental concept in geometry. It is expressed as \((x-h)^2 + (y-k)^2 = r^2\). This equation represents a circle on the Cartesian coordinate plane. The terms within the equation provide essential information about the circle, such as its center and radius.

• \((h, k)\) represent the coordinates of the circle’s center.
• \(r\) represents the radius of the circle.

The standard form is derived from the distance formula, which calculates the distance between a fixed point, the center, and any point on the circle which is equal to the radius. It ensures that all points \((x, y)\) that satisfy the equation lie exactly on the circle.

Center of a Circle

Locating the center of a circle is crucial for understanding its position in the coordinate system. In the standard form equation \((x-h)^2 + (y-k)^2 = r^2\), the values \(h\) and \(k\) offer the coordinates of the circle's center.

• The center is given as \((h, k)\).
• By observing the expression \((x-h)\), we determine the center's x-coordinate as \(-h\) when rewritten as \(x - (-h)\).
• Similarly, for \((y-k)\), the y-coordinate is \(-k\) identified by \(y - (-k)\).

For example, in the equation \((x+8)^2 + (y-5)^2 = 13\), the center is calculated as \((-8, 5)\). Thus, any symmetric property of the circle on a graph will revolve around this midpoint.

Radius of a Circle

The radius of a circle defines the distance from its center to any point on its boundary. In the standard form equation \((x-h)^2 + (y-k)^2 = r^2\), the radius is represented by \(r\). Once squared, it becomes \(r^2\), indicating the radius in the equation.

• If you have \(r^2 = 13\), then \(r = \sqrt{13}\).

The radius is vital not only in determining the size of the circle but also in solving problems related to circumference, area, and spatial positioning. It directly influences how large or small the circle appears relative to its center.

Point on a Circle

Determining whether a point lies on a circle involves substitution and calculation. You plug the point’s x and y values into the circle's equation. If both sides of the equation match, the point is on the circle.

For example, to test if \((-5, 2)\) is on the circle \((x+8)^2 + (y-5)^2 = 13\):

• Substitute \(x = -5\) and \(y = 2\).
• Compute: \((-5 + 8)^2 + (2 - 5)^2 = 9 + 9 = 18\).
• Since 18 does not equal 13, the point \((-5, 2)\) does not lie on the circle.

This method ensures that only specific points, where the calculated distance matches the radius, are acknowledged to lie on the circle’s boundary.