In this article we will solve exercises on the equation of the circle which exemplify how to find said equation given prior information about the circle. For example, given the center, radius, diameter, points of intersection with the axes, tangency to the axes, etc. Also, we will deal with cases where given the equation of the circle we need to extract precise information about it such as its center, radius, points of intersection with the axes, tangency, etc.

The following solved exercises will help us deepen our understanding of the elements of the circle and will test our knowledge about it.

1

Write the equation of the circle with the center and radius

Solution

1. We substitute the data into the standard equation of the circle:



where:


are the coordinates of the center and is the radius.


2. We expand the powers and obtain:

 

2

Write the equation of the circle with the center and radius

Solution

1. We substitute the data into the standard equation of the circle:



where:


are the coordinates of the center and is the radius.


2. We expand the powers and obtain:

 

3

Write the equation of the circle with the center and radius

Solution

1. We substitute the data into the standard equation of the circle:

 

 

where:

 

are the coordinates of the center and is the radius.

 

2. We expand the powers and obtain:

 

4

Write the equation of the circle with the center and diameter

Solution

1. We substitute the data into the standard equation of the circle:

where:

are the coordinates of the center and is the radius.

2. We expand the powers and obtain:

 

5

Calculate the equation of the circle whose center is at and passes through the point

Solution

1. We substitute the data into the standard equation of the circle:

where:

are the coordinates of the center and is the radius.

2. We substitute, expand the powers and obtain the radius:

 

 

3. We substitute the value of the radius and the center, expand the powers and obtain:

 

6

Calculate the equation of the circle whose center is at and is tangent to the axis

Solution

1. Since the circle is tangent to the axis, then the radius equals the distance from the center to the axis. Therefore,

2. We substitute the value of the radius and the center into the general equation of the circle, expand the powers and obtain:

 

7

Calculate the equation of the circle whose center is at and is tangent to the axis

Solution

1. Since the circle is tangent to the axis, then the radius equals the distance from the center to the axis. Therefore,

2. We substitute the value of the radius and the center into the general equation of the circle, expand the powers and obtain:

 

8

Calculate the equation of the circle whose center is at and is tangent to the horizontal line

Solution

1. Since the circle is tangent to the line , then the radius equals the distance from the center to said tangent line. Therefore,

2. We substitute the value of the radius and the center into the general equation of the circle, expand the powers and obtain:

 

9

Calculate the equation of the circle whose center is at and is tangent to the vertical line

Solution

1. Since the circle is tangent to the line , then the radius equals the distance from the center to said tangent line. Therefore,

2. We substitute the value of the radius and the center into the general equation of the circle, expand the powers and obtain:

 

10

Calculate the equation of the circle that has a diameter with endpoints and

Solution

1. Since we know the endpoints of the diameter, then the midpoint is the center of the circle:

2. To find the radius, we substitute the value of the center and one of the endpoints of the diameter into the general equation of the circle:

 

 

3. We substitute the value of the radius and the center into the general equation of the circle, expand the powers and obtain:

 

11

Given the circle with equation , find the center and radius.

Solution

1. We rewrite the equation ordering the and and complete the perfect square trinomials:

2. We factor the perfect square trinomials:

 

 

y

12

Determine the coordinates of the center and radius of the circles:

A

B

C

D

Solution

A

We rewrite the equation in its standard form:

 

 

y

 

B

 

y

 

Since is imaginary, it is not a real circle.

 

C

 

Dividing by 4 and rewriting the equation in standard form:

 

 

 

y

 

D

 

Dividing by 4 and rewriting the equation in standard form:

 

 

 

y

13

Calculate the equation of the circle whose center is at and is tangent to the x-axis

Solution

1. We graph the circle with the given data:


representacion grafica de circunferencia con centro en 2, -3 y tangente al eje de abscisas

 

2. From the graph we can deduce that:

         

 

 

14

Calculate the equation of the circle whose center is at and is tangent to the y-axis

Solution

1. We graph the circle with the given data:

 

representación gráfica de circunferencia tangente al eje de ordenadas, con centro en -2, 8

 

2. From the graph we can deduce that:

         

 

 

15

Calculate the equation of the circle whose center is at the intersection point of the lines , , and whose radius equals

Solution

1. We set up a system of equations with the given lines; the solution of the system corresponds to the center of the circle:

 

         

 

2. We substitute and into the standard form:

 

 

 

representacion grafica de circulo con centro en -1, 0

16

Find the equation of the circle concentric with the equation , and that passes through the point

Solution

1. Since they are concentric they have the same center:

 

representacion grafica de circulos concentricos con centro en 3 y -1

 

2. We calculate the center of the circle :

 

 

 

3. To calculate the radius we calculate the distance from to

 

 

4. We substitute the center and radius into the standard form:

 

 

17

Find the equation of the circle whose center is at point and is tangent to the line:

Solution

1. The radius is calculated with the distance from point to the line :

2. We substitute and into the standard form:

 

 

 

representacion grafica de circulo con centro 3 y 1, y con recta tangente

18

Find the equation of the circle that passes through the points

Solution

1. Considering the general equation of a circle as , we substitute the given points and construct a system of equations:

2. We solve the system of equations and substitute into the general form considered:

 

                   

 

19

Find the equation of the circle circumscribed to the triangle with vertices:

Solution

representación gráfica de la circunferencia circunscrita al triángulo

 

1. Considering that the vertices of the triangle are points through which the circle passes, we can consider the equation of the circle as and substitute the given points:

2. We solve the system of equations and substitute into the general form considered:

 

 

                   

 

20

Find the equation of the circle that passes through points and and has its center on the line:

Solution

representacion grafica de circunferencia con centro en una recta

 

1. Let's consider that point is the center of the circle and is on the line ; we can set up the system:

 

 

2. From the first 2 equations we obtain:

 

 

3. Solving the system:

 

                 

 

21

Calculate the equation of the circle that passes through point , whose radius is and whose center lies on the bisector of the first and third quadrants

Solution

representacion gráfica de circulo con centro en la recta

 

1. Let's consider that point is the center of the circle; moreover, the bisector of the first and third quadrants is the line :

 

 

 

         

 

 

 

2. We obtain 2 solutions for :

 

                                           

 

3. For :

 

 

 

 

 

 

4. For :

 

 

 

 

22

The endpoints of the diameter of a circle are points and . What is the equation of this circle?

Solution

representación gráfica de un circulo y una recta con dos puntos en la circunferencia A y B y con centro C

 

1. The radius of the circle will be half the distance between points and :

 

 

2. The center of the circle will be at the midpoint between points and :

 

 

3. We obtain the coefficients and for the form :

 

         

 

 

23

Find the equation of the circle concentric with the circle that is tangent to the line

Solution

representación gráfica de circunferencia concéntrica a la circunferencia con recta tangente

 

1. We obtain the center of the circle with coordinates :

 

         

 

         

 

2. The radius will be the distance between and the line :

 

 

3. We obtain the coefficients and for the form :

 

                 

 

24

Calculate the relative position of the circle and the line

Solution

representacion de posicion relativa de una recta en una circunferencia

 

1. We set up a system of equations between the equation of the circle and the equation of the line to find their intersections:

 

         

 

 

 

 

         

 

         

 

Since there are two points of intersection, we can say that the line and the circle are secant.

25

Study the relative position of the circle with the lines:

A

B

C

Solution

A

 

representacion gráfica de una recta y un circunferencia secantes

We set up a system of equations between the equation of the circle and the equation of the line to find their intersections:

 

 

         

 

       

 

       

 

Since there are two points of intersection, we can say that the line and the circle are secant.

 

B

 

grafica de circulo y recta tangente

 

We set up a system of equations between the equation of the circle and the equation of the line to find their intersections:

 

 

   

 

                   

 

Since there is only one point of intersection between the circle and the line, we can say they are tangent.

 

C

 

dibujo de circulo y recta no tangente

 

We set up a system of equations between the equation of the circle and the equation of the line to find their intersections:

 

 

         

 

 

Since there are no points of intersection between the line and the circle, we can say they are exterior.

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.