1. Find the centre and radius of each of the following circle,(x-1)² + y² =4.
2. Find the equation of the circle whose centre is (1, 2) and which passes through the point (4, 6).
3. Find the equation of a circle which touches x-axis at a distance 5 from the origin and radius 6 units.
4. Find the equation of circle passing through the origin, radius 17 and ordinate of the centre is -15.
5. Find the equation of the circle which touches the axes and whose centre lies on x-2y=3.
6. A circle whose centre is the point of intersection of the lines2x-3y+4=0 and3x+4y-5=0 passes through the origin. Find its equation.
7. One diameter of the circle circumscribing the rectangle ABCD is 4 y = x+7 . If the coordinates of A and B are (-3, 4) and (5, 4) respectively, find the equation of the circle.
8. Find the equation of the circle whose centre is (3,-1) and which cut-off an intercept of length 6 from the line 2x-5y+18=0.
9. Find the equations of the circles touching y-axis at (0, 3) and making an intercept of 8 units in the X-axis.
10.Find the equation of the circle whose centre is at (3, 4) and which touches the line5x+12=1 .
11.Find the equation of the circles passing through two points on Y-axis at distance 3 from the origin and having radius 5.
12. If the line 2x-y+1=0 touches the circle at the point (2, 5) and the centre of the circle lies on the lines x+y-9=0 . Find the equation of the circle.
13.Find the coordinates of the centre and radius of the following circle: 2x²+ 2y²-3x + 5y = 7
14.Find the coordinates of the centre and radius of the following circle: ½ (x²+y²)+ x cos + y sin -4 = 0
15.Find the equations of the circle passing through the points : (5,7),(8,1) and (1,3)
16.Find the equations of the circle passing through the points : (1,2),(3,-4) and (5,-6)
17. Find the equations of the circle which passes through the points (3,7), (5,5) and has its centre on the line x-4y = 1.
18.Find the equations of circle which passes through the origin and cuts off chords of lengths 4 and 6 on the positive sides of the x-axis and y-axis respectively.
19.Find the equations of the circle concentric with x2+ y2 4x-6y-3 = 0 and which touches the y-axis.
20.Find the equation of the circle, the end point of whose diameter are (2,-3) and (-2, 4). Find its centre and radius.
21.Find the equation of a circle which touches y-axis at a distance of 4 units from the origin and radius 3 units;
22.Show that the point (x, y) where lies on a circle for all values ofq.
23.Find the Cartesian equation of the following curves. Wherever the curve is a circle, find its centre and radius
24.Find the equation of the circle when the end points of a diameter are (5,-3) and (2,-4).
25.Find the equation of the circle when the end points of a diameter are (p, q) and (r, r).
26.Find the equation of the circle drawn on a diagonal of the rectangle as its diameter whose sides are given by x = 5, x = 8; y = 4, y = 7;
27.Find the equation of the circle drawn on the intercept made by the line 2x + 3y = 6 between the co-ordinate axes as diameter.
28.Find the equation of the circle which has (1, 3) and (4, 5) as opposite ends of a diameter. Find also the equation of the perpendicular diameter.
29.Find the equation of the circle passing through the origin and the points where the line meets the axes of co-ordinates.
30.Find the equation of the circle which circumscribes the triangle formed by the lines x = 0, y = 0 and ax + by = 1.
31.Find the equation of the circle which passes through the origin and makes positive intercepts 3 and 4 units on the x-axis and y-axis respectively.
32.Examine whether the points (2, 2), (1, 5) and (-1,-3) lie inside, outside or on the circle.
33.Examine whether the point (1, 2) and (2, 1) lie on the same side or opposite sides of the circle.
34.Find the co-ordinates of the points of intersection of the line and the circle
35.Find the points of intersection of the line and the circle
36.For what values of c will the line be a tangent to the circle
37.Prove that the line touches the circle Also, find the point of contact.
38.Find the equations of two tangents to the circle which make an angle of 600with the x-axis.
39.Find the length of the chord intercepted by the circle and the line .
40.Find the equations of the tangents to the circle which are parallel to the line
41.Find the equation of the tangents to the circle which are perpendicular to the line .
42.Find the equation of the circle passing through the (0, 0) and cutting intersects a and b on the positive side of x and y axes respectively.
43.Find the equation of the circle passing through the origin and cutting intercepts 10 and 24 from the positive sides of x and y axis respectively.
44.Find the equation of the circle touching :9 (i) x-axis and having centre at (4, 3) (ii) x-axis at the origin an having radius 10
45.Find the equation of the circle having radius 5 and passing through two points on x-axis at distances 4 from the origin.
46.Find the equation of image of the circle in the x-axis.
47.Find the equation of the circle passing through the point (2, 4) and centre at the point of intersection of the lines and .
48.Find the equation of the circle which passes through the point or intersection of and and whose centre is (2, 3).
49.Find the equation of the circle passing through the centre of the circle and being concentric with the circle .
50.Find the equation of the circle whose centre is (1,-3) and which touches the line .
51.Find the equation of the circle whose radius is 5 and the centre lies onthe positive side of x-axis at a distance 5 from the origin.
52.Find the equation of the circle which passes through the points (-1, 2) and (3,-2) and whose centre lies on the line .
53.Find the equation of the circle which touches the axis of y at a distance 4 from the origin and cuts off an intercept of length 6 on the axis of x.
54.Does the point (-2.5, 3.5) lie inside outside or on the circle ?
55.Find the centre and radius of each of the following circles. (i) (ii)
56.Find the centre and radius of each of the following circle. (i) (ii)
57.Find the centre and radius of the following circles (i) (ii)
58.Find the equation of the circles passing through the following three points. (1,-2), (5, 4) and (10, 5)
59.Determine whether the following equations represent a circle or not
60.Determine whether the following equations represent a circle, a point or no circle
61.Find the parametric equations of the following circles: 62.Find the equation of a circle with centre (h, k) and touching the x-axis. 63.Find the equation of a circle with centre (h, k) and touching the y-axis. 64.Find the equation of a circle with centre (h, k) and touching both the axes. 65.Find the equation of a circle with centre (2, 4) and radius 5 66.Find the equation of a circle with centre (-3,-2) and radius 6 67.Find the equation of a circle with centre (a, a) and radius . 68.Find the equation of a circle with centre and radiusa. 69.Find the equation of a circle with centre and radius 8 . 70.Find the equation of a circle with centre at the origin and radius 4 71.Find the centre and radius of each of the following circles: 72.Find the centre and radius of each of the following circles: 73.Find the equation of the circle whose centre is (2,-5) and which passes through the point (3, 2). 74.Find the equation of the circle of radius 5 cm, whose centre lies on the y-axis and which passes through the point (3, 2). 75. If two diameters of a circle lie along the lines and , and the area of the circle is 38.5 sq cm, find the equation of the circle.
76.Show that the equation represents a circle. Also, find its centre and radius.
77.Show that the equation represents a circle.
78.Show that the equation represents a circle. Find its centre and radius.
79.Show that the equation represents a circle. Find the centre and radius.
80.Show that the equation represents a circle. Find its centre and radius.
81.Show that the equation represents a point circle.
82.Show that the equation does not represent a circle.
83.Find the equation of the circle which is circumscribed about the triangle whose vertices are A(-2, 3), B(5, 2) and C(6,-1). Find the centre and radius of this circle.
84.Find the equation of the circle passing through (1, 1), (2, 2) and having its radius unity. Show that there exists two such circles.
85.Find the equation of the circle passing through the points (2, 3) and (-1, 1) and whose centre is on the line .
86.Find the equation of circle which passes through (2,-2) and (3, 4) and whose centre lies on the line 87.Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line . 88.Find the equation of the circle whose centre is (2,-3) and radius is 8. 89.Find the equation of circle which touches both the coordinate axes and the line x = 4. 90.Find the equation of a circle which touches y-axis at (0, 4) and cuts off intercept of 6units on x-axis. 91.Find the centre and radius for each of the following circles :C (i) (ii) (iii) (iv) (v) 92.Find the equation of a circle concentric with the circle .
93.Find the equation of circumcircle of a triangle whose vertices are (1, 0), (-1, 0) and (0, 1). 94.Find the equation of a circle concentric with the circle and with three times its radius. 95.Prove that the points (-1, 6), (5, 2), (7, 0) and (-1,-4) are not con cyclic. 96.Show that the points (4, 3), (8,-3) and (0, 9) do not lie on a circle. 97.Find the equation of a circle having line segment, with ends points (0,-1) and (2, 3), as diameter. 98.Find the centre and radius of the circles . 99.Find the centre and radius of the circle. 100. Find the equation of a circle with centre with centre (2, 2) and passes through the point (4, 5). 101. Find the equation of the circle which passes through the points (20, 3), (19, 8) and (2,-9). Find its centre and radius. 102. If the line touches the circle , then find the value of k. 103. Find the equation of the circle with (i) Centre (-2, 3) and radius 4 (ii) Centre (a, b) and radius (iii) Centre (0,-1) and radius 1. 104. Find the centre and radius of each of the following circle, . 105. Find the equation of the circle passing through the point of intersection of the lines and and whose centre is the point of intersection of the lines and 106. If the equations of two diameters of acircle are and and the radius is 10, find the equation of the circle. 107. A circle of radius 4 units touches the coordinate axes in the first quadrant. Find the equations of its images with respect to the line mirrors
108. The circle is rolled along the positive direction of x-axis and makes one complete roll. Find its equation in new-position. 109. Show that the points (3,-2), (1,0),(-1,-2) and (1,-4) are concyclic 110. Find the equations of the circle which circumscribes the triangle formed by the lines . x + y + 3 = 0, x-y + 1 = 0 and x = 3 111. Find the equations of the circle which circumscribes the triangle formed by the lines . 2x + y-3 = 0, x + y-1 = 0 112. Prove that the centres of the three circles x2+ y2-4x-6 y-12 = 0. x2+ y2+ 2x + 4y-10 = 0 and x2+ y2-10x 16 y-1 = 0 are collinear. 113. Prove that the radii of the circles x2+ y2= 1 , x2+ y2-2x-6y-6 = 0 and x2+ y2-4x + 12 y-9 = 0 114. Find the equation of the circle the end points of whose diameter are the centres of the circles and . 115. The sides of a square are , , and . Find the equation of a circle drawn on the diagonal of the same square as its diameter. 116. Find the equation of the circle circumscribing the rectangle whose sides are and 117. The abscissa of the two points A and B are the roots of the equation and their ordinates are the roots of the equation . Find the equation of the circle with AB as diameter. Also, find its radius. 118. ABCD is a square whose side is a ; taking AB and AD as axes, prove that the equation of the circle circumscribing the square is . 119. The line meets the circle at A and B. Find the equation of the circle on AB as diameter.
120. Find the equation of the circle whose diameter is the line segment joining (-4, 3) and (12,-1). Find also the intercept made by it on y-axis.
121. Find the equation of the circle concentric with the circle and having its area equal to 16psquare units. 122. Find theequation of the circle whose centre is at the point (4, 5) and passes through the centre of the circle . 123. Show that the equations x = a cosq+ b sinqand y = a sinq-b cosqrepresents a circle,qbeing a parameter. 124. Show that the point (x, y) given by lie on a circle for all real values of t. 125. Eliminate the parameter ‘t’ from the equations 126. Find the equation of the circle passing through the points (0, 0), (-3, 0) and (0, 8) 127. Find the equation of the circle on the straight line joining the points of intersection of and lx + my = 1 as diameter. 128. The abscissa of the two points A and B are the roots of the equation and their ordinates are the roots of the equation x2+ x-6 = 0. Find the equation of the circle with AB as diameter. 129. On the line joining (1, 0) and (3, 0) an equilateral triangle is drawn, having its vertex in the first quadrant. Find the equation to the circles described on its sides as diameter. 130. Find the lengths of the intercepts of the following circle on the x-axis; 131. Find the lengths of the intercepts of the following circle on the y-axis. 132. Find the equation of a circle passing through the points (1, 2) and (3, 0) and cuts off an intercept of 4 units on the x-axis. 133. Find the equation of the circle touching the y-axis at a distance of-3 units from the origin and cuts off an intercept of 8 units on the x-axis.
134. Prove that the line touches the circle . Also find the point of contact.
135. Show that the line touches the circles and at the same point. 136. Prove that the line touches the circle at the point 137. Find the co-ordinates ofthe point of intersection of the line and the circle . Also, find the length of the chord segment. 138. Find the length of the intercept on the straight line by the circle 139. If the line lx + my+ n = 0 touches the circle then prove that 140. Find the co-ordinates of the middle point of the chord which the circle cuts off on the line . 141. Prove that the centres of the circles . and are collinear. 142. Prove that the radii of the circle and are in arithmetic progression. 143. Find the equation of the circle which is concentric with the and the double of its area. 144. Find the equation of a circle concentric with the circle and of double its area. 145. Find the equation of the circle concentric with the circle and having radius double of its radius.
146. Find the equation of the circle concentric with the circle and passing through the point (-4,-5). 147. The line is tangent at the point (-3, 0) and the line is tangent at the point (4, 1) to a circle. Find the equation of the circle. 148. Find the equation of the circle passing through the points (0,-1) and (2, 0) and whose centre lies on the line . 149. Find the equation of the circle passing through the points(2,-3) and (3,-2) and whose centre lies on the line . 150. Find the equation of the circle drawn on the intercept between the axes made by the line as a diameter. 151. The abscissa of two points A and B are the roots of the equation and the ordinates are the roots of the equation . Find the equation of the circle with AB as its diameter. Also find the coordinates of the centre and the length of the radius of the circle. 152. If (4, 1) be an end of a diameter of the circle. , find the coordinates of the other end of the diameter. 153. Find the equation of the circle circumscribing the rectangle whose sides are 154. Find the cartesian equation of the following curves whose parametric equations are: (i) (ii) (iii) 155. Prove that and are the parametric equations of a circle for all satisfying .
156. Show that and . Represent a circle where is the parameter.
157. Show that the point (x, y) where Lies on a circle for all values of . 158. Show that the point (x, y), where Lies on a circle for all values of . 159. Find the equation of a circle of radius 5 units, whose centre lies on the x-axis and which passes through the point (2, 3). 160. If isa chord of the circle , find the equation of the circle with this chord as a diameter. 161. Find the equation of the circle whose centre is (2,-3) and which passes through the intersection of the line and 162. Find the equation of the circle passing through the point (-1, 3) and having its centre at the point of intersection of the lines and 163. Find the equation of the circle, the coordinates of the end points of one of whose diameters are (i) A(-2,-3) and B(-3, 5) (ii) A(p, q) and B(r, s) 164. The sides of a rectangle are given by the equations and Find the equation of the circle drawn on the diagonal of this rectangle as its diameter. 165. Find the equation of a circle passing through the points (5, 7), (6, 6) and (2,-2). Find its centre and radius. 166. Find the equation of the circle passing through the vertices of a triangle whose sides are presented by the equations and 167. Show that the points (5, 5), (6, 4), (-2, 4) and (7, 1) are concyclic, i.e., all lie on the same circle. Find the equation, centre and radius of this circle. 168. Find the equation of the circle concentric with the circle and passing through the point P(5, 4). 169. Show that the points A(1, 0), B(2,-7), C(8, 1) and D(9,-6) all lie on the same circle. Find the equation of this circle, its centre and radius.
170. Find the equation of the circle which passes through the points (1, 3) and (2,-1), and has its centre on the line
171. Find the equation of the circle concentric with the circle and which touches the y-axis. 172. Find the equation of the circle concentric with the circle and of double its area. 173. Prove that the centres of the three circles , and are collinear. 174. Find the equation of the circle which passes through the points A(1, 1) and B(2, 2) and whose radius is 1. Show that there are two such circles. 175. Find the equation of a circle passing through the origin and intercepting lengths a and b on the axes. 176. Find the equation of the circle circumscribing the triangle formed by the lines and 177. Show that the quadrilateral formed by the straight lines , and is cycle and hence find the equation of the circle. 178. If (-1, 3) and are the extremities of the diameter of the circle , find the coordinates . 179. Show that the line touches the circle . 180. Find the equation of the circle which passes through the centre of the circle and is concentric with the circle . 181. Find the equation of the circle which passes through the origin and cuts off intercepts 3 and 4 on positive part of x-axis and y-axis respectively. 182. Show that the line touches the circle . Find the point of contact.
183. Find the equation of the circle passing through the points (0,-1) and (2, 0) and whose centre lies on the line
184. Find the equation of the circle whose centre is on the line and which passes through (3,-2) and (-2, 0).
185.
186. Find the equation of a circle whose centre is (3,-2) and which passes through the intersection of the lines and .
187. Find the equation of a circle whose centre is a point (1,-2) and which passes through the centre of the circle
188. Find the equation of a circle which passing through the points (-3, 1), (6, 4), (2,-6).
189. Find the equation of circumcircle of a triangle whose vertices are (1, 0), (-1, 0), and (0, 1)
190. Find the equation of a circle concentric with the circle and with three times its radius.
191. Line meets the axes at A and B find the equation of the circle through O, A, B.
192. The equations of two diameters of a circle, of radius 7 ,units, are and . Find the equation of the circle.
193. Find the equation of the circle concentric with circle and double its area.
194. Find the equation of the circle passing through the points (2, 3) and (-1, 1) and whose centre is on the line
195. Find the equation of a circle concentric with the circle and has double of its area.
