Problem 53 - Precalculus

Question

The center of a circle is the point \((3,2) .\) If the point (-2,-10) lies on this circle, find the standard equation for the circle.

Step-by-Step Solution

Answer

The standard equation is \((x - 3)^2 + (y - 2)^2 = 169\).

Step 1: Identify the Circle's Components

The standard 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. Here, the center of the circle is given as \((3, 2)\), so \(h = 3\) and \(k = 2\).

Step 2: Use the Distance Formula to Find the Radius

A point \((-2, -10)\) lies on the circle, so it can be used to determine the circle's radius. Use the distance formula to find the radius: \[ r = \sqrt{(-2 - 3)^2 + (-10 - 2)^2} \] Simplify the expression:\[ r = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] Thus, the radius \(r = 13\).

Step 3: Write the Standard Equation of the Circle

Plug \(h = 3\), \(k = 2\), and \(r = 13\) into the standard equation format:\[(x - 3)^2 + (y - 2)^2 = 13^2\]This becomes:\[(x - 3)^2 + (y - 2)^2 = 169\]

Key Concepts

Distance Formula in Geometry

In geometry, the distance formula helps us calculate the distance between two points in a coordinate plane. This formula is derived from the Pythagorean theorem and is written as: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. The formula effectively gives us the length of the hypotenuse of a right triangle when we consider \((x_2 - x_1)\) and \((y_2 - y_1)\) as the lengths of the other sides.

This formula is crucial in various geometric problems, especially when determining the radius of a circle. Knowing the distance between a circle's center and a point on its circumference helps validate a standard equation of a circle.

• To use the formula: Identify your two points.
• Subtract the x-coordinates, \((x_2 - x_1)\), to find the horizontal distance.
• Subtract the y-coordinates, \((y_2 - y_1)\), to find the vertical distance.
• Square these values, sum them, and take the square root to find the total distance.

Center of a Circle

The center of a circle is crucial in determining its properties and equations. It is the point from which every point on the circle is equidistant. In the standard equation of a circle \((x - h)^2 + (y - k)^2 = r^2\), the point \((h, k)\) represents the center. Understanding this concept is important for constructing and manipulating circle equations.

The center is not just an abstract concept; it's practically used in various aspects of geometry:

• It helps define the circle's position on the coordinate plane.
• It is essential for understanding the transformation of circles in terms of translations and reflections.

When given a circle's center, you have half the information needed to form the circle's equation. Combine it with the radius, and you've got your equation!

Radius of a Circle

The radius is a vital part of understanding and working with circles. It is defined as the distance from the circle's center to any point on its circumference. The radius is constant for a given circle and is symbolized by \(r\) in mathematical formulas.

To find the radius from a given point on the circle and its center, use the distance formula, which acts as a bridge between geometry and algebra. This enables us to solve circle problems more efficiently:

• Calculate the distance between the center \((h, k)\) and any given point on the circle, say \((x_1, y_1)\).
• The result will be the radius \(r\), which is crucial for writing the circle's standard equation.

Beyond merely calculating the radius, this measurement allows for a deeper understanding of the circle's size and how it occupies space in a plane. It helps explain and predict the circle's interactions with other shapes and lines, such as tangents and secants.