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Equation of the circle: Course, method and corrected exercises

This article aims to present how to calculate the equation of a circle and recognize which circle it is, through the course, examples and corrected exercises.

Definition

The Cartesian equation of the circle in a plane is written in the form:

(x-x_A)^2 + (y-y_A)^2 = R^2 

with:

  • (xA,yA) the center of the circle
  • R the radius of the circle

So if we know the radius of the circle and its center, it is easy to establish its Cartesian equation

Corrected exercises and methods

Find the equation of the circle from its center of its radius

We have the following statement: Consider the circle of radius 2 and radius (1,3). Find the equation of this circle. We have, according to the definition that the equation is written:

(x-1)^2 + (y-3) ^2 = 2^2

We will then develop this equation to simplify it:

x^2 -2x +1 +y^2 -6y +9 = 4

Then, we will simplify and put all the elements on the left:

x^2 +y^2 -2x-6y +6 = 0

We have therefore found the equation of the circle with center (1,3) and radius 2.

Find the circle from its equation

Here is the typical statement:

Find the circle associated with the following equation:

x^2 - 2x + y^2 +6y = 0 

For this, we will use the canonical form for the terms in x:

\begin{array}{ll} x^2 - 2 x & = x^2 - 2x +1 - 1 \\ &= (x-1)^2 -1 \end{array}

Then those in there:

\begin{array}{ll} y^2 +6 x & = y^2 +6y +9 - 9 \\ &= (y+3)^2 -9 \end{array}

If we combine the terms, we now have:

 (x-1)^2 -1 + (y+3)^2 -9 = 0

We pass the constants on the right:

 (x-1)^2 + (y+3)^2 = 10

Which we rewrite in the form:

 (x-1)^2 + (y+3)^2 = (\sqrt{10})^2

It is therefore the circle with center (1,-3) and radius √10

Check that a point belongs to a circle

Statement: Let the circle of equation

(x-2)^2+(y-4)^2 = 25

Does the point (0;5) belong to the circle?

To do this, we replace x and y with the coordinates of our point. We then obtain:

(0-2)^2 +(5-4)^2 = 5 \neq 25

So the point does not belong to the circle because the left member is not equal to 25.

Exercices

Exercise 1

Write the equation of the circle from the coordinates of the center Ω​ and the radius R:

\begin{array}{l} 1)\ \Omega = (1;6) , R= 4\\ 2)\ \Omega = (5;3) , R= 2\\ 3)\ \Omega = (7 ;4) , R= 6\\ \end{array}

Exercise 2

From the diameter [AB], give the equation of the circle:

\begin{array}{l} 1)\ A= (1;6) , B=(2;3)\\ 2)\ A= (2;5) , B=(3;2)\\ 3) \ A= (4;5) , B=(1;1)\\ \end{array}

Exercise 3

Write the equation of the circle from the coordinates of the center Ω​ and a point A:

\begin{array}{l} 1)\ \Omega = (1;6) , A= (1;3)\\ 2)\ \Omega = (2;4) , A= (3;1)\\ 3)\ \Omega = (4;2) , A= (5;5)\\ \end{array}

Exercise 4

In an orthonormal reference of the plane, determine the coordinates of the center and the radius of the circle whose equation is:

x^2+y^2 -12x + 4y -3 = 0 

Exercise 5

In an orthonormal reference of the plane, determine the coordinates of the center and the radius of the circle whose equation is:

\begin{array}{l} 1)\ x^2+y^2 = 0\\ 2)\ x^2 + y^2 = - 1\\ 3)\ (x-3)(x+2) +(y-5)(y+6)=0 \end{array}

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