Problem 7 - Technical Mathematics with Calculus

Question

Find the center and radius of each circle. Graph. $$x^{2}+y^{2}=49$$

Step-by-Step Solution

Answer

The center of the circle is \( (0, 0) \) and the radius is 7 units. Graph this by drawing a circle with the given radius centered at the origin on a coordinate plane.

Step 1: Identify the General Form of a Circle

The general form of 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. Compare this with the given equation \(x^2 + y^2 = 49\).

Step 2: Determine the Center of the Circle

In the given equation \(x^2 + y^2 = 49\), the \(x\) and \(y\) terms are not shifted, which means \(h = 0\) and \(k = 0\). The center of the circle is thus at the origin, \( (0, 0) \).

Step 3: Calculate the Radius of the Circle

The constant term on the right side of the equation, \(49\), is equal to the square of the radius \(r^2\). Taking the square root of both sides gives \(r = \sqrt{49} = 7\). The radius of the circle is 7 units.

Step 4: Plot the Circle on a Graph

Draw a coordinate plane and mark the center of the circle at \( (0, 0) \). Use a compass to draw a circle with a radius of 7 units around the center. The circle should be drawn such that all points on the circumference are 7 units away from the origin in all directions.

Key Concepts

General Form of a Circle

When it comes to understanding the general form of a circle's equation, it's like decoding a secret message about a perfect round shape on a flat surface. The general equation for any circle is \(x-h)^2 + (y-k)^2 = r^2\), where \(h, k\) represents the circle's center, and \(r\) stands for the circle's radius. This formula is based on the Pythagorean Theorem, which expresses the relationship between the sides in a right triangle and finds its way into the realm of circles to define their boundary on a graph.

Think of \(h\) and \(k\) as GPS coordinates, guiding us to the circle's central meeting point, and the \(r\) as a consistent measure from this center to any point along the curve. In our exercise, \(x^2 + y^2 = 49\), the equation looks slightly different because our circle's center \( (0,0) \) is at the very heart of the coordinate plane, making this an exceptionally symmetrical situation where \(h = 0\) and \(k = 0\).

Circle Center and Radius

Picturing a circle's center and radius is not just about drawing dots and lines; it's grasping the core (literally) of the circle's structure. The circle's center (\(h, k\)) is the anchor point from which every edge of the circle is equidistant. It's like the bullseye on a dartboard. In our exercise, identifying the center couldn't be simpler as it's positioned at \( (0,0) \) - the origin of the graph where the x and y axes cross paths.

Now, the radius - that's the circle's halfway marker from the center to any point on the circumference. It's the magician's wand that, with a single length measurement (in our case, 7 units), draws the entire circle into existence. By squaring the radius and setting it equal to 49, we deciphered that the circle's radius is the square root of 49, which is 7. This vital number holds the clue to the circle's size: the larger the radius, the bigger the circle.

Plotting Circles in Coordinate Plane

Setting a circle onto the graph paper's grid is an art that starts with a single point: the center. After pinning down our center at \( (0,0) \) for the exercise, we step into the shoes of an explorer, compass in hand (the geometric kind, not the magnetic one). We stretch our compass legs to reach the radius length of 7 units and twirl around the center, gently guiding the pencil to leave a trace that curves back to the start.

To ensure our circle isn't a lopsided pancake or a squashed balloon, every plotted point must maintain an unwavering arm's length from the center, spanning out uniformly in all directions. That's the circle's democratic rule—every point on the edge is equally important and equally distant from the hub. This equality is what gives the circle its harmonious round shape, a shape that's celebrated across mathematics, design, and nature.