Introduction
A circle is a simple closed curve that is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the center. The distance between any point of the circle and the center is always the same, so the circle is said to be symmetrical about its center. The circumference of a circle is the length of one complete revolution along the edge of the circle, and it is related to the radius by $$C=2\pi r$$ In this lesson, we will learn how to find the equation of such a circle when we are given its center and one point on the curve.
The Basics of the Equation of a Circle
A circle is a two-dimensional shape that is defined by a center point and a radius. The equation of a circle is used to determine the properties of a circle, such as its circumference, diameter, and area. The equation of a circle can also be used to find the equation of a tangent line to a circle.
The Center
To find the equation of a circle with a center at (7, 2) and a point on the circle at (2, 5), we can use the standard form of the equation of a circle. This form is:
(x – h)^2 + (y – k)^2 = r^2
where (h, k) is the coordinates of the center of the circle, and r is the radius.
A Point on the Circle
In order to find the equation of a circle with a center at (7,2) and a point on the circle at (2,5), we will need to use the distance formula. The distance formula is given by:
d=|(x_1-x_2)^2+(y_1-y_2)^2|
In this equation, (x_1,y_1) is the coordinates of the center of the circle and (x_2,y_2) is the coordinates of a point on the circumference of the circle. Applying this formula to our situation, we get:
d=|(7-2)^2+(2-5)^2|
d=|5^2+(-3)^2|
d=5^2+9
d=25+9
d=34
How to Find the Equation of a Circle with a Center at (7, 2) and a Point on the Circle at (2, 5)
There are a few steps that you need to take in order to find the equation of a circle with a center at (7, 2) and a point on the circle at (2, 5). You will need to use the Pythagorean theorem and some basic algebra. We will walk you through each step so that you can get the answer that you are looking for.
Substitute the Values into the Formula
To find the equation of a circle with a center at (7, 2) and a point on the circle at (2, 5), we need to substitute the values into the formula for the equation of a circle. The formula is:
(x-h)^2+(y-k)^2=r^2
where (h,k) is the center of the circle, and r is the radius. In our case, h=7 and k=2. We can calculate r by using the distance formula:
r=(x_1-x_2)^2+(y_1-y_2)^2
where (x_1, y_1) and (x_2, y_2) are points on the circle. In our case, (x_1, y_1)=(7, 2) and (x_2, y_2)=(2, 5). Therefore:
r=(7-2)^2+(2-5)^=(5-(-3))^2=(8-5)^2=9
Therefore, the equation of our circle is:
(x-7)+(y-2)=9
Solve for “r”
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. We know the center of the circle is at (7,2) and a point on the circle is at (2,5). We can plug those values in for x, y, h and k to solve for r.
(x-h)^2 + (y-k)^2 = r^2
(2-7)^2 + (5-2)^2 = r^2
(-5)^2 + 3^2 = r^2
25 + 9 = r^ 2
34 = r^ 2
r= √34
r= 5.83
Write the Final Equation
Write the final equation in standard form: (x-center)^2 + (y-center)^2= radius^2
In this case, the center is (7,2) and the point on the circle is (2,5), so the equation becomes:
(x-7)^2 + (y-2)^2= radius^2
Conclusion
Now that we know what the equation of a circle is and how to find the center and radius of a circle, we are ready to find the equation of a circle with a center at 7 2 and a point on the circle at 2 5.
First, we need to find the radius of the circle. We can do this by using the distance formula:
r = √((x2-x1)²+(y2-y1)²)
r = √((2-7)²+(5-2)²)
r = √((-5)²+(3)²)
r = √(25+9)
r = √(34)
r ≈ 5.811
Thus, the radius of the circle is 5.811. Now that we know the radius, we can plug it into the equation of a circle to find the equation of our specific circle:
(x-7)²+(y-2)²=5.811