Note: The general equation of circle is where the center is and radius

Question 1: Find the coordinates of the center and radius of each of the following circles :

i)      ii)

iii)      iv)

Answer:

i)        Give equation :

Comparing it with , we get,

Therefore center

Radius

ii)      Give equation :

Comparing it with , we get,

Therefore center

Radius

iii)      Given equation

Comparing it with , we get,

Therefore center

Radius

iv)     Given equation

Comparing it with , we get,

Therefore center

Radius

Question 2: Find the equation of the circle passing through the points:

(i) and      (ii) ,and

(iii) and      (iv) and

Answer:

(i)      The circle passes through the points and

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Substituting in i) we get

     … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

(ii)     The circle passes through the points ,and

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Substituting in i) we get

     … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

(iii)    The circle passes through the points and

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Substituting in i) we get

     … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

(iv)    The circle passes through the points and

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Substituting in i) we get

     … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

Question 3: Find the equation of the circle which passes through and has its center on the line .

Answer:

The circle passes through and has its center on the line .

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Since center lie on , we get

     … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

Question 4: Find the equation of the circle which passes through the points and has its center on the line .

Answer:

The circle passes through and has its center on the line .

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Since center lie on , we get

     … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

Question 5: Show that the points and are concyclic.

Answer:

Consider and

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Substituting in i) we get

     … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

     … … … … … v)

Now we check if satisfies equation  v)

. Therefore S lies on the circle.

Therefore, and are con-cyclic.

Question 6: Show that the points and all lie on a circle, and find its equation, center and radius.

Answer:

Consider and

The equation of circle is      … … … … … i)

Substituting in i) we get

     … … … … … ii)

Substituting in i) we get

     … … … … … iii)

Substituting in i) we get

   … … … … … iv)

Solving ii) , iii)  and iv) simultaneously, we get

Therefore the equation of the required circle is:

     … … … … … v)

Now we check if satisfies equation  v)

. Therefore S lies on the circle.

Therefore, and are con-cyclic.

Therefore center

Radius units.

Question 7: Find the equation of the circle which circumscribes the triangle formed by the lines:

i) and      

ii) and

iii)   and      iv)   and

Answer:

i)        Given equations:

     … … … … … i)

     … … … … … ii)

     … … … … … iii)

Solving i) , ii) and iii) we get the vertices

The equation of circle is      … … … … … iv)

Substituting in iv) we get

     … … … … … v)

Substituting in iv) we get

     … … … … … vi)

Substituting in iv) we get

   … … … … … vii)

Solving v) , vi)  and vii) simultaneously, we get

Therefore the equation of the required circle is:

ii)      Given equations:

     … … … … … i)

     … … … … … ii)

     … … … … … iii)

Solving i) , ii) and iii) we get the vertices

The equation of circle is      … … … … … iv)

Substituting in iv) we get

     … … … … … v)

Substituting in iv) we get

     … … … … … vi)

Substituting