Problem 48 - Algebra 2

Question

Use the given information to write an equation of the circle. center \((1,-2),\) through \((0,1)\)

Step-by-Step Solution

Answer

The equation of the circle is \((x-1)^{2}+(y+2)^2=9\).

Step 1: Calculate the Radius

The radius (r) can be determined by using the formula to find the distance between two points. The formula is \(r=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\). Substituting (h,k)=(1,-2) as the center of the circle and (x1,y1)=(0,1) in the formula, we get \(r=\sqrt{(0-1)^{2}+(1-(-2))^{2}}=\sqrt{9}=3\)

Step 2: Formulate the Equation

Now, we plug the obtained radius and the given center into the standard equation for the circle. Substituting h=1, k=-2 and r=3 into the equation \((x-h)^{2}+(y-k)^2=r^{2}\), we get \((x-1)^{2}+(y+2)^2=3^{2}\)

Key Concepts

Radius Calculation

To calculate the radius of a circle when you have its center and a point on the circle, you can use the distance formula. This formula helps to find the distance between two points in a coordinate plane. Here, our center is at point \( (1, -2) \,\) and the point on the circle is at \( (0, 1) \,\). You enter these into the distance formula to compute the radius.

The distance formula is \( r = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \,\). It's like a modified version of the Pythagorean theorem, where you calculate the horizontal and vertical distances between two points and then find the hypotenuse. For our example, plug in the numbers: \( r = \sqrt{(0-1)^2 + (1 - (-2))^2}\,\). Simplifying, you get: \( r = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} = 3\,\).

This means the radius of the circle is 3 units. Can you imagine this circle on a graph? The center is at \( (1, -2) \,\), and every point on the circle is exactly 3 units away. It's like drawing a perfect curve equidistant from the center point.

Distance Formula

The distance formula is a handy tool in geometry to determine the distance between any two points in a plane. It's particularly useful when dealing with circles. This is because a circle can be defined as the set of all points equidistant from a single point, called the center. Here’s how the distance formula comes into play:

Consider two points, \( (x_1, y_1) \,\) and \( (x_2, y_2) \,\). The distance \( d \,\) between them can be found with: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \,\). This formula calculates how far apart these two points are on the coordinate plane.

• Subtract the x-coordinates: \( (x_2 - x_1) \,\)
• Subtract the y-coordinates: \( (y_2 - y_1) \,\)
• Square both differences, add them together, and take the square root of the result.

This not only helps in finding the radius when we know a point on the circle and the center, but it also comes in handy in numerous applications, such as finding the shortest path between locations.

Standard Equation of a Circle

The standard equation of a circle is a crucial part of understanding and writing equations for circles. When given a circle's center and its radius, the standard form of the equation is \( (x-h)^2 + (y-k)^2 = r^2 \,\).

Here, \( (h, k) \,\) is the center of the circle, and \( r \,\) is its radius. This shows that every point \( (x, y) \,\) on the circle is importantly \( r \,\) units away from the center. Let's break down the process:

• Identify the coordinates of the center of the circle \( (h, k) \,\). For instance, in our example, it's \( (1, -2) \,\).
• Determine the radius \( r \,\). In our example, it is 3 units.
• Substitute these values into the equation. With \( h = 1, k = -2 \,\) and \( r = 3 \,\), the circle's equation becomes: \( (x-1)^2 + (y+2)^2 = 9 \,\).

By using the equation this way, you can clearly see the relationship between the center and any point on the circle, reminding you of the constant radius throughout the entire shape. This fundamental property will aid in visualizing and solving many geometric problems.