# 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=2x^{2}+y^{2}+3 x+12 y+2=0

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(3,7),(5,5)x-4 y=1x^{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)

*** QuickLaTeX cannot compile formula: The center *** Error message: Missing $ inserted.

(-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