EXERCISE 24.2
QUESTION 1
Find the co-ordinates of the center and radius of each of the following circles:
(i)
Sol :
The given equation can be rewritten as
Center
And, radius
= 7
(ii)
Sol :
The given equation can be rewritten as
Center
Radius
(iii)
Sol :
The given equation can be rewritten as
Center
radius
= 3
(iv)
Sol :
The given equation can be rewritten as
Center
radius
QUESTION 2
Find the equation of the circle passing through the points:
(i) and
Sol :
Let the required circle be
It passes through and
Substituting the coordinates of these points in equation (1) :
Simplifying (2),(3) and (4)
Equation of the required circle:
(ii) and
Sol :
Let the required circle be
It passes through and
Substituting the coordinates of these points in equation (1) :
Simplifying (2),(3) and (4)
The equation of the required circle is
(iii) and
Sol :
Let the required circle be (1)
It passes through and
Substituting the coordinates of these points in equation (1):
(3)
Simplifying and
The equation of the required circle is
(iv) and
Sol :
Let the required circle be
It passes through and
Substituting the coordinates of these points in equation (1):
Simplifying (2),(3) and (4)
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
Sol
Let the required equation of the circle be (1)
It is given that the circle passes through
The center lies on the line
Solving (2),(3) and g=\frac{3}{2}f=6c=2
x^{2}+y^{2}+3 x+12 y+2=0
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(3,7),(5,5)x-4 y=1
x^{2}+y^{2}+2 g x+2 f y+c=0 . \ldots(1)
(3,7),(5,5)
58+6 g+14 f+c=0 \ldots(2)
50+10 g+10 f+c=0 \quad \ldots(3)
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(-g,-f)x-4 y=1
-g+4 f-1=0 \quad \ldots .(4)
(4) :
Hence, the required equation of the circle is
QUESTION 5
Show that the points and
are concyclic.
Sol :
Let the required equation of the circle be
It is given that the circle passes through
Solving (2),(3) and (4)
Therefore, the equation of the circle is
We see that the point satisfies the equation (5)
Hence, the points and
are concyclic.
QUESTION 6
Show that the points and
all lie on a circle, and find its equation, center and radius.
Sol :
Let the required equation of the circle be
It is given that the circle passes through
Solving (3) and (4) by subtracting ,we get
then substituting value of g in equation (2) and (3) we get
Solving (5) and (6) by subtracting , we get
then substituting value of f in equation (6) , we get
Thus, the equation of the circle is
We see that the point satisfies equation (7)
Hence, the points and
lie on the circle.
Also, center of the required circle
Radius of the required circle
QUESTION 7
Find the equation of the circle which circumscribes the triangle formed by the lines :
(i) and
Sol :
In
Let represent the line
Let BC represent the line
Let CA represent the line
Let A,B,C are the points of intersection of lines (i) and (ii) , (ii) and (iii) , (iii) and (i) respectively
are the co-ordinates
Now A circle
Circle passes through
Solving (1),(2),(3) we get
Hence, the required equation of the circumcircle is
(ii) and
Sol :
In
Let AB represent the line
Let BC represent the line
Let CA represent the line
Intersection point of and
is
Intersection point of and
is
Intersection point of and
is
The coordinates of and
are
and
respectively.
Let the equation of the circumcircle be
It passes through and
Hence, the required equation of the circumcircle is
(iii) and
Sol :
In
Let AB represent the line
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Let BC represent the line
Let CA represent the line (3)
Intersection point of (1) and (3) is
Intersection point of (1) and (2) is
Intersection point of (2) and (3) is
The coordinates of and
are
and
respectively.
Let the equation of the circumcircle be
It passes through A,B and C
Hence, the required equation of the circumcircle is
(iv) and
Sol :
In
Let represent the line
Let BC represent the line
Let CA represent the line
Intersection point of (1) and (3) is
Intersection point of (1) and (2) is
Intersection point of (2) and (3) is
Therefore, the coordinates of and
are
and
respectively.
Let the equation of the circumcircle be
It passes through and
Hence, the required equation of the circumcircle is
QUESTION 8
Prove that the centers of the three circles and
are collinear.
Sol :
The given equations of the circles are as follows:
The center of circle (1) is
The center of circle (2) is
The center of circle (3) is
The area of the triangle formed by the points and
is
= 0
Hence, the centers of the circles and
are collinear.
QUESTION 9
Prove that the radii of the circles and
are in A.P.
Sol :
Let the radii of the circles ,
and
be
and
,
,
Now ,
and
are in A.P
QUESTION 10
Find the equation of the circle which passes through the origin and cuts off intercept of lengths 4
and 6 on the positive side of the x-axis and y-axis respectively.
Sol :
According to the question, the circle passes through the origin.
Let the equation of the circle be
The circle cuts off chords of lengths 4 and 6 on the positive sides of the x-axis and the y- axis, respectively.
Center
Required equation :
QUESTION 11
Find the equation of the circle concentric with the circle and double of its area.
Sol :
Let the equation of the required circle be
The center of the circle is
Area of the required circle
Here, radius of the given circle
Area of the required circle
Let R be the radius of the required circle.
Thus, the equation of the required circle is i.e.
QUESTION 12
Find the equation to the circle which passes through the points and whose radius is 1 .Show that there are two such circles.
Sol :
Let the equation of the required circle be
It passes through and
From (1) and (2), we have:
From and
we have:
by using
Solving (4) and (5) , we get
Therefore, the required equations of the circles are and
Hence, there are two such circles.
QUESTION 13
Find the equation of the circle concentric with and which touches the y-axis.
Sol :
since, the circles are concentric Centre of required circle = Centre of
The centre of the required circle is
We know that if a circle with centre touches the
-axis, then
is the radius of the circle.
Thus, the radius is 2
Equation of the circle:
QUESTION 14
If a circle passes through the point
then find the coordinates of its center .
Sol :
The general equation of the circle is
Now, it is passing through
Also, it is passing through
Again, it is passing through
The coordinates of its centre are given by
QUESTION 15
Find the equation of the circle which passes through the points and
and the center lies on the straight line
Sol :
The general equation of the circle is where the center of the circle is
Now, it is passing through
Also, it is passing through
Now, the center lies on the straight line ,
satisfies this equation
Solving (1),(2) and (3), we get
and
The equation of the circle is given by