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15.    Does the point (–2.5, 3.5) lie inside, outside or on the circle x^2 + y^2 = 25?

Given, a circle   As we can see center of the circle is ( 0,0) Now the distance between (0,0) and (–2.5, 3.5) is  Since distance between the given point and center of the circle is less than radius of the circle, the point lie inside the circle.

14.    Find the equation of a circle with centre (2,2) and passes through the point (4,5).

Let the equation of circle be : Now, since the centre of the circle is (2,2), our equation becomes Now, Since this passes through the point (4,5) Hence  The Final equation of the circle becomes

13.  Find the equation of the circle passing through (0,0) and making intercepts a and b on the coordinate axes.

Let the equation of circle be, Now since this circle passes through (0,0) Now, this circle makes an intercept of a and b on the coordinate axes.it means circle passes through the point (a,0) and (0,b) So, Since  Similarly, Since b is not equal to zero. So Final equation of the Circle ;

12.    Find the equation of the circle with radius 5 whose centre lies on x-axis and passes through the point (2,3).

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; So let the circle be, Since it's radius is 5 and its centre lies on x-axis, And Since it passes through the point  (2,3). When  ,The equation of the circle is : When  The equation of the circle is :

11.  Find the equation of the circle passing through the points (2,3) and (–1,1) and hose centre is on the line x - 3y - 11 = 0.

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; Given Here, Condition 1: the circle passes through points   (2,3) and (–1,1) Here,  Now, Condition 2: centre is on the line. From condition 1 and condition 2  Now let's substitute this value of h and k in condition 1 to find out  So now, the Final Equation of the circle is 

10.  Find the equation of the circle passing through the points (4,1) and (6,5) and whose centre is on the line 4x + y = 16.

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; Given Here, Condition 1: the circle passes through  points (4,1) and (6,5) Here,  Now, Condition 2:centre is on the line . From condition 1 and condition 2  Now lets substitute this value of h and k in condition 1 to find out r  So now, the Final Equation of the circle is 

9. Find the centre and radius of the circles.

       2x^2 + 2y^2 - x = 0

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; Given here  Can also be written in the form  So, from comparing, we can see that  Hence Center of the circle is the Radius of the circle is   .

8.  Find the centre and radius of the circles.

      x^2 + y^2 -8x +10y -12 = 0

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; Given here  Can also be written in the form  So, from comparing, we can see that  Hence the radius of the circle is .

7.  Find the centre and radius of the circles.

      x^2 + y^2 -4x - 8y - 45 = 0

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; Given here  Can also be written in the form  So, from comparing, we can see that  Hence the Radius of the circle is .

6. Find the centre and radius of the circles.

       (x+5)^2 + (y-3)^2 = 36

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; Given here  Can also be written in the form  So, from comparing, we can see that  Hence the Radius of the circle is 6.

5. Find the equation of the circle with

       centre (-a,-b) and radius \sqrt{a^2 - b^2}

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; So Given Here   AND  So the equation of the circle is: , 

4.Find the equation of the circle with

     centre (1,1) and radius \sqrt2

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; So Given Here   AND  So the equation of the circle is: , 

3.  Find the equation of the circle with

   centre \left(\frac{1}{2},\frac{1}{4} \right ) and radius \frac{1}{12}

 

As we know,  The equation of the circle with center ( h, k) and radius r is give by ; So Given Here   AND    So the equation of circle is: , 

2. Find the equation of the circle with

    centre (–2,3) and radius 4

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; So Given Here   AND  So the equation of the circle is: , 

1. Find the equation of the circle with

    centre (0,2) and radius 2

As we know,  The equation of the circle with centre ( h, k) and radius r is given by ; So Given Here   AND  So the equation of the circle is: , 
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