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

Question 1: Find the equation of the circle with:

(i) Centre and radius (ii) Centre and radius

(iii) Centre and radius (iv) Centre and radius

(v) Centre and radius

Answer:

(i) Given: Centre and radius

Therefore the equation of the circle is:

(ii) Given: Centre and radius

Therefore the equation of the circle is:

(iii) Given: Centre and radius

Therefore the equation of the circle is:

(iv) Given: Centre and radius

Therefore the equation of the circle is:

(v) Given: Centre and radius

Therefore the equation of the circle is:

Question 2: Find the center and radius of each of the following circles:

i) ii)

iii) iv)

Answer:

i) Given equation

Comparing it with general equation of circle where is the center of the circle and is the radius of the circle.

Center and Radius

ii) Given equation

Comparing it with general equation of circle where is the center of the circle and is the radius of the circle.

Center and Radius

iii) Given equation

Comparing it with general equation of circle where is the center of the circle and is the radius of the circle.

Center and Radius

iv) Given equation

Center and Radius

Question 3: Find the equation of the circle whose center is and which passes through the point .

Answer:

Given circle whose center is and which passes through the point

Radius

Therefore the equation of the circle is

Question 4: Find the equation of the circle passing through the point of intersection of the lines and and whose center is the point of intersection of the lines and .

Answer:

The center of the circle is the point of intersection of lines and which is

The circle passes through the point of intersection of lines and which is

Therefore the Radius

Therefore the equation of the circle is :

Question 5: Find the equation of the circle whose center lies on the positive direction of y-axis at a distance from the origin and whose radius is .

Answer:

Given center and Radius

Therefore the equation of the circle is :

Question 6: If the equations of two diameters of a circle are and and the radius is , find the equation of the circle.

Answer:

Center of the circle is the point of intersection of lines and which is

Therefore the equation of the circle is :

Question 7: Find the equation of a circle

(i) which touches both the axes at a distance of units from the origin.

(ii) which touches x-axis at a distance from the origin and radius units

(iii) which touches both the axes and passes through the point

iv) passing through the origin , radius and ordinate of the center is

Answer:

Let be the center of the circle with radius . Thus the equation will be

(i) It is given that the circle passes through the points and

… … … … … i)

Also

… … … … … ii)

From i) and ii)

From ii) we get ,

, since

Therefore

Hence the equation of the circle is:

(ii) It is given that the circle with radius units touch the x-axis at a distance of units from origin.

Therefore center is

Hence the equation of the required circle is:

(iii) It is given that the circle touches both the axes.

Thus the required equation will be:

Also the circle passes through the point

Therefore

Hence the required equations are:

or

iv) Given

The equation passes through the point

Therefore the equation of the circle is:

Hence the required equations of the circle are

Question 8: Find the equation of the circle which has its center at the point and touches the straight line .

Answer:

Given the center of the circle

Perpendicular distance of from the line is

Therefore the equation of the circle will be

Question 9: Find the equation of the circle which touches the axes and whose center lies on .

Answer:

Let the circle touches and on the axes. Therefore the center is and radius is .

Since the center is lies on , we get

Therefore the center is and radius

Therefore the equation of the circle is

Question 10: A circle whose center is the point of intersection of the lines and passes through the origin. Find its equation.

Answer:

The intersection point of and is given by

The circle passes through , therefore

Radius

Therefore the equation of the circle is:

Question 11: A circle of radius units touches the coordinate axes in the first quadrant. Find the equations of its images with respect to the line mirrors and .

Answer:

Given circle has a radius of and touches the coordinate axes in first quadrant.

Therefore the center

The image of on is . Therefore the equation of the circle is

The image of on is . Therefore the equation of the circle is

Question 12: Find the equations of the circles touching y-axis at and making an intercept of units on the x-axis.

Answer:

Case 1: Center lies in 1st Quadrant

The circle touches y-axis at and makes an intercept of units on x-axis

Therefore

Let the required line be

In

Therefore

Therefore the coordinate of center and Radius

Therefore equation of the circle is:

Case 2: Center lies in 2nd Quadrant

Therefore the coordinate of center and Radius

Therefore equation of the circle is:

Question 13: Find the equations of the circles passing through two points on y-axis at distances from the origin and having radius .

Answer:

The circle passes through and and Radius

Let the center of the circle be . Therefore the equation of the circle is:

… … … … … i)

Substituting in equation i) we get

… … … … … ii)

Substituting in equation i) we get

… … … … … iii)

Solving ii) and iii) we get

Therefore

Hence the equation of the required circle is

Question 14: If the lines and are the diameters of a circle of area square units, then obtain the equation of the circle.

Answer:

The point of intersection of and is

Therefore center

Given Area of the circle sq. units

Therefore the equation of the circle is

Question 15: If the line touches the circle , then find the value of .

Answer:

The center of the circle is and Radius

Perpendicular distance from the center to the tangent is equal to the radius

Question 16: Find the equation of the circle having as its center and passing through the intersection of the lines and .

Answer:

Point of intersection of and is .

Center of the circle is .

Therefore the radius

Therefore the equation of the circle is:

Question 17: If the lines and are tangents to a circle, then find the radius of the circle.

Answer:

Given lines:

Therefore the two lines are parallel to each other. The distance between the two lines is

Therefore the radius of the circle is units.

Question 18: Show that the point given by and lies on a circle for all real values of such that , where is any given real number.

Answer:

Given: and

Squaring and adding we get:

Hence the above equation represents the equation of the circle on which lies.

Question 19: The circle is rolled along the positive direction of x-axis and makes one complete roll. Find its equation in new-position.

Answer:

The equation of the circle is:

Therefore center is and Radius

Distance covered in one roll .

Therefore the center would move towards right direction.

Therefore the new center would be

Hence the new equation of the circle is:

Question 20: One diameter of the circle circumscribing the rectangle is . If the coordinates of and are and respectively, find the equation of the circle.

Answer:

Refer to the adjoining diagram.

Given and

Slope of

Mid point of

The equation of the perpendicular bisector

The point of intersection of and is

Therefore the center is

Radius

Therefore the equation of the circle is:

Question 21: If the line touches the circle at the point and the center of the circle lies on the line . Find the equation of the circle.

Answer:

Refer to the adjoining diagram.

Let the center of the circle be

Therefore the equation of the circle is:

The circle passes through , therefore

Substituting , we get

Therefore the equation of the circle is :