Problem 31 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(0,0), r=7 $$

Step-by-Step Solution

Answer

The standard form of the equation of the circle with center at the origin (0,0) and radius 7 is \(x^2 + y^2 = 49\).

Step 1: Identify values

Given in the exercise, the center of the circle is at the origin (0,0) and the radius \(r\) is 7.

Step 2: Apply values to standard form

Substitute the given center coordinates and radius into the standard form of a circle equation which is \((x-a)^2 + (y-b)^2 = r^2\). Since the center is (0,0), \(a = 0\) and \(b = 0\). The radius \(r = 7\). Hence the circle equation becomes \(x^2 + y^2 = 49\).

Key Concepts

Standard Form

In geometry, the standard form of a circle's equation is a compact way to express all circles on a coordinate plane. This form provides vital details such as the center and radius at a glance. The standard form of the equation for a circle is \((x-a)^2 + (y-b)^2 = r^2\). Here,

• \(a\) and \(b\) denote the coordinates of the center of the circle \((a,b)\).
• \(r\) is the radius of the circle.

This format ensures that with the values of \(a\), \(b\), and \(r\), anyone can quickly understand the circle's location and size. The square terms \((x-a)^2\) and \((y-b)^2\) result from applying the Pythagorean theorem to points on the circle and the center.

Center of a Circle

The center of a circle is a crucial element as it determines the position of the circle on a plane. The point \((a, b)\) in the equation \((x-a)^2 + (y-b)^2 = r^2\) identifies the exact center. In practical terms, it tells us how far the circle is from the origin of the coordinate plane's axes.Understanding the center is vital when graphing the circle, as it acts as the midpoint from which the radius extends in all directions. The center does not have to be at the origin; it can be at any point on the plane, allowing diverse placement of the circle.

Radius of a Circle

The radius of a circle is the fixed distance from its center to any point on the perimeter of the circle. In the standard form equation, it appears as \(r^2\). Thus, if you know \(r^2\), you find the radius \(r\) by taking the square root. For example, if the equation is \(x^2 + y^2 = 49\), the radius is \(r = \sqrt{49} = 7\). The significance of the radius lies in its consistent length from the center, which dictates the circle's size. A larger radius means a bigger circle, and vice versa. The radius is always positive.

Origin

The origin is the point \((0, 0)\) on a coordinate plane where the x-axis and y-axis intersect. It's the foundational point of reference in graphing. When a circle's center is at the origin, the equation simplifies beautifully into a more basic form because both \(a\) and \(b\) are zero. This gives us the equation \(x^2 + y^2 = r^2\) for a circle centered at the origin. This simplicity aids significantly in visualization and calculations. Points directly on the axes have immediate relationships with the origin since the horizontal and vertical distances directly reflect the radius.