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