Problem 5 - The Calculus with Analytic Geometry

Question

In Exercises 5 through 10, find an equation of the circle satisfying the given conditions. Center is at \((1,2)\) and through the point \((3,-1)\).

Step-by-Step Solution

Answer

(x - 1)^2 + (y - 2)^2 = 13

Step 1 - Identify the standard form of a circle's equation

The standard form of a circle's equation is given by ewline ewline \[(x - h)^2 + (y - k)^2 = r^2\] ewline ewline where \(h, k\) is the center of the circle and \(r\) is the radius.

Step 2 - Substitute the center coordinates \(h, k\)

Substitute the center of the circle, \((1, 2)\) into the equation. This modifies the equation into ewline ewline \[(x - 1)^2 + (y - 2)^2 = r^2\]

Step 3 - Use the given point to find the radius

Substitute the point \((3, -1)\) into the equation to find the radius. ewline ewline \[(3 - 1)^2 + (-1 - 2)^2 = r^2 \] which simplifies to ewline ewline \[2^2 + (-3)^2 = r^2 ewline \rightarrow 4 + 9 = r^2 ewline \rightarrow r^2 = 13\]

Step 4 - Write the final equation

Substitute \text{r}^2 back into the circle's equation to get the final form: ewline ewline \[(x - 1)^2 + (y - 2)^2 = 13\]

Key Concepts

standard form of a circle's equation

Understanding the standard form of a circle's equation is crucial for solving problems related to circles. The standard form is given by the formula:


The expressions The values

(x - h)^2 + (y - k)^2 = r^2

If you break it down:

• (h, k) - These are the coordinates of the circle's center.
• r - This is the radius of the circle.

So, if you know the center and the radius of a circle, you can easily plug these values into the formula to get the equation of the circle. This formula is very useful in geometry and various applications in mathematics.

Now, let's see this concept in action with an example:
Suppose you have a circle with the center at (1, 2) and it passes through the point (3, -1). How do we find the equation of this circle?We'll start by using our standard form equation mentioned above.

radius of a circle

Finding the radius of a circle is an essential step in determining its equation. The radius is the distance from the circle's center to any point on the outer edge of the circle. You might have heard the term 'radius' in the context of different shapes or objects. In this example, we'll use a point on the circle to calculate the radius.

From the given problem, we have a circle centered at (1, 2) and passing through the point (3, -1). To find the radius, we use these steps:

• Calculate the horizontal distance: 3 - 1 equals 2.
• Calculate the vertical distance: -1 - 2 equals -3.
• Apply the Pythagorean theorem:

by squaring and adding the distances:
Calculate:
< (2)^2 + (-3)^2 = r^2
Once you have r^2,You can proceed to write the final equation.

coordinates of the center

Understanding the coordinates of the center of a circle is fundamental in forming its equation. The coordinates are represented as (h, k), where:

• h - This is the x-coordinate of the center.
• k - This is the y-coordinate of the center.

In the given problem, the center of the circle is at (1, 2). So, here the values for h and k are 1 and 2, respectively.

To write the equation of the circle:

Substitute

equation

result in: < >(x - 1)^2 + (y - 2)^2 < = r^2 <

Finally, write the equation file complete by swapping the calculated value of r²:
ej(x - 1)^2 + (y - 2)^2 = 13

<1, ><2 > and < passing through the point > <3><-1>
< understanding how to substitute the coordinates of the center into the standard form of a circle's equation helps solve such problems quickly and accurately.