( 1 ) Obtain the equation of the circle passing through the points ( 5, - 8 ), ( - 2, 9 ) and ( 2, 1 ).

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1 PROBLEMS 03 CIRCLE Page ( ) Obtain the equation of the irle passing through the points ( 5 8 ) ( 9 ) and ( ). [ Ans: x y 6x 48y 85 = 0 ] ( ) Find the equation of the irumsribed irle of the triangle formed by three lines given by x y = 6 x y = 4 and x y = 5. [ Ans: x y 7x 9y 50 = 0 ] ( 3 ) Find entre and radius of the irle whose equation is 4x 4y x 4y 9 = 0. Ans : 3 3 ( 4 ) Find the equation of the irle touhing both the axes and passing through ( ). [ Ans: x y x y = 0 x y 0x 0y 5 = 0 ] ( 5 ) The artesian equation of the irle is x y 4x y 4 = 0. Find its parametri equations. [ Ans: x = 3 os θ y = 3 sin θ θ ( π π ] ] ( 6 ) Show that the linesegments joining any point of a semiirle to the end points of the diameter are perpendiular to eah other. ( 7 ) Show that the point ( 4 5 ) is inside the irle x y 4x 6y 5 = 0. Find the point on the irle whih is at the shortest distane from that point. [ Ans: ( 5 6 ) ] ( 8 ) Find the length of the hord of the irle x y 4x y 0 = 0 ut off by the line x y 0 = 0. [ Ans: ]

2 PROBLEMS 03 CIRCLE Page ( 9 ) Find the midpoint of the hord of the irle x y = 6 ut off by the line x 3y 3 = 0. [ Ans: ( 3 ) ] ( 0 ) Find the onditions for the line x os α y sin α = p to be the tangent to the irle x y = r. [ Ans: p = r ] ( ) Find the equations of the tangents to the irle x y = 7 from the point ( 5 3 ). [ Ans: 4x y 7 = 0 x 4y 7 = 0 ] ( ) The lengths of the tangents drawn from a point P to two irles with entre at origin are inversely proportional to the orresponding radii. Show that all suh points P lie on a irle with entre at origin. ( 3 ) Find the measure of an angle between two tangents to the irle x y = a drawn from the point ( h k ). [ Ans: tan ( a / h k a ) ] ( 4 ) Find the set of all points P outside a irle x y = a suh that the tangents to the irle drawn from P are perpendiular to eah other. [ Ans: x y = a ] ( 5 ) Find the equation of the irle whih passes through the points of intersetion of the irles x y = 3 and x y x y 4 = 0 and whose entre lies on the line 4x y 6 = 0. [ Ans: x y 4x 4y 9 = 0 ] ( 6 ) Show that the irles x y ax = 0 and x y by = 0 are orthogonal to eah other.

3 PROBLEMS 03 CIRCLE Page 3 ( 7 ) If the points A ( a 0 ) A ( a 0 ) B ( 0 b ) and B ( 0 b ) are on a irle then prove that aa = bb. Also find the equation of the irle. [ Ans: x y ( a a )x ( b b )y aa = 0 ] ( 8 ) Show that the point of intersetion of the lines given by x 5xy y 7x 5y 3 = 0 with the axes lie on a irle. Find its equation. [ Ans: x y 7x 5y 3 = 0 ] ( 9 ) Find the equation of the irle whose diametrially opposite points are the points of intersetion of the line y = mx with the irle x y ax = 0. [ Ans: ( m ) ( x y ) a ( x my ) = 0 ] ( 0 ) Find the set of the midpoints of the hords of the irle x y = a formed by the line passing through ( x y ). [ Ans: S = { ( x y ) l x y x x y y = 0 and x y < a } ] ( ) Find the equation of the irle whih is orthogonal to the irles x y 6x = 0 and x y 4y = 0 and the entre of whih lies on the line 3x 4y 6 = 0. [ Ans: 3x 3y 4x 6y 5 = 0 ] ( ) If the entre of the irle passing through the origin and orthogonal to the irle x y 4x y 4 = 0 lies on the line x y 4 = 0 then find the equation of the irle. [ Ans: x y 4x 4y = 0 ] ( 3 ) The irle orthogonal to the irles x y 6x 9 = 0 and x y x y 7 = 0 passes through the point ( 0 ). Find its equation. [ Ans: x y 4x 8y 3 = 0 ] ( 4 ) If the irles x y gx fy = 0 and x y g x f y = 0 are tangent to eah other then show that f g = fg.

4 PROBLEMS 03 CIRCLE Page 4 ( 5 ) If the line x os α y sin α = p ontains a hord of the irle x y = a then find the equation of the irle whose diameter is this hord. [ Ans: x y a p ( x os α y sin α p ) = 0 ] ( 6 ) Find the equation of the irle whih passes through ( ) ( 4 3 ) and whih has a diameter along the line 3x 4y = 7. [ Ans: 5x 5y 94x 8y 55 = 0 ] ( 7 ) Find the equation of the irumirle of a triangle with verties ( a b ) ( a b ) and ( a b a b ) ( b 0 ) [ Ans: b ( x y ) ( a b ) x ( a b ) ( a b ) = 0 ] ( 8 ) Find the length of the ommon hord of ( x a ) y = a and x ( y b ) = b a b > 0. Also find the equation of the irle with this ommon hord as diameter. Ans : ab a b ( a b ) ( x y ) ab ( bx ay ) = 0 ( 9 ) Find the length of the ommon hord of ( x a ) ( y b ) = and ( x b ) ( y a ) =. [ Ans: 4 ( a b ) ] ( 30 ) If the irles x y gx a = 0 and x y fy a = 0 touh eah other then establish that g f = a. ( 3 ) Find the equation of the irle with the ommon hord of the irles x y 4x = 0 and x y 6y = 0 as a diameter. [ Ans: 3x 3y 36x 4y = 0 ] ( 3 ) Prove that the two irles x y ax by a b = 0 and x y bx ay a b = 0 interset eah other at right angles.

5 PROBLEMS 03 CIRCLE Page 5 ( 33 ) Find the equation of the irle passing through ( ) touhing the Xaxis and having its entre on the line x y = 3 in the first quadrant. [ Ans: x y 4x y 4 = 0 ] ( 34 ) Find the equations of the irles touhing both the oordinate axes and also the line 3x 4y 6 = 0. [ Ans: ( ) x y 6x 6y 9 = 0 ( ) 4x 4y 4x 4y = 0 ( 3 ) x y x y = 0 ( 4 ) 4x 4y x y 9 = 0 ] ( 35 ) Find the equation of the irle passing through ( 0 ) and touhing the lines x y = 0 and x y 8 = 0. [ Ans: ( x 5 ) ( y ) = 0 ( 5x 9 ) (5 y ) = 500 ] ( 36 ) Find the equation of the ommon hord of the two irles x y ax by = 0 and x y bx ay = 0 ( a b 0 ). Using this equation derive the ondition for the two irles to touh eah other. [ Ans: x y = 0 ( a b ) 8 = 0 ] ( 37 ) Find the equation of the irle whih has as a diameter the segment ut off by the irle x y gx fy = 0 on the line l x my n = 0. [ Ans: ( l m ) ( x y gx fy ) ( n mf gl ) ( l x my n ) = 0 ] ( 38 ) If A and B are the points of ontat of the tangents drawn from P ( 3 4 ) to the irle x y = 6 find the area of triangle PAB. [ Ans: 08 / 5 ] ( 39 ) Prove that the length of the ommon hord of the two irles x y = a and ( x ) y = b 4 is where = area of a triangle having sides of lengths a b and.

6 PROBLEMS 03 CIRCLE Page 6 ( 40 ) Find the equations of the irles passing through ( 4 3 ) and touhing the lines x y = and x y =. [ Ans: x y ( )x ( ) = 0 and x y ( )x ( ) = 0 ] ( 4 ) Prove that the length of the ommon hord of the two irles x y = a and ( x ) y = b is [ ( a b ) ( a b ) ( a b ) ( a b ) ]. ( 4 ) If m i m i > 0 i = 3 4 are four distint points on a irle then show mi that m m m 3 m 4 =. ( 43 ) A irle passes through ( a b ) and touhes the Xaxis. Prove that the lous of the other end of the diameter of the irle through ( a b ) is ( x a ) = 4by. ( 44 ) A is the entre of the irle x y x 4y 0 = 0. The tangents at the points B ( 7 ) and D ( 4 ) on the irle meet at the point C. Find the area of the quadrilateral ABCD. [ Ans: 75 sq. units ] ( 45 ) Show that the two irles whih pass through the points ( 0 a ) and ( 0 a ) and touh the line y = mx ut eah other orthogonally if = a ( m ). ( 46 ) Prove that if l 3m 4l = 0 then the line l x my = 0 touhes a fixed irle and find the equation of the fixed irle. [ Ans: x y 4x = 0 ] ( 47 ) Find the equation of the irle of minimum radius passing through ( 3 ) and touhing the irle x y 9x y 5 = 0. [ Ans: x y 5x 0y 5 = 0 ]

7 PROBLEMS 03 CIRCLE Page 7 ( 48 ) Using oordinate geometry prove that the angles subtended by any hord of a irle at any two points on its irumferene on the same side of the hord are equal. ( 49 ) Using oordinate geometry prove that the angle subtended by any hord of a irle at the entre of the irle is twie the angle subtended by the same hord at any point on the irumferene on the same side as the entre. ( 50 ) The tangents drawn from an external point P ( x y ) to the irle x y gx ay = 0 touhes the irle at the points A and B. Find the equation of the irle passing through the points P A and B. [ Ans: x y ( g x )x ( f y )y g x f y = 0 ] ( 5 ) The tangents drawn from an external point P ( x y ) to the irle x y gx ay = 0 touhes the irle at the points A and B. O is the entre of the irle. Find the area of ( i ) triangle PAB ( ii ) triangle OAB and ( iii ) quadrilateral OAPB. Ans : ( i ) ( ii ) ( iii ) g gx f fy ( x y gx fy ) ( x g ) ( y ( g f ) ( x y gx fy ) ( x g ) ( g f ) ( x y gx fy ) f ) ( y f ) 3 ( 5 ) A line through an external point P ( x y ) intersets the irle x y gx ay = 0 in A and B. If PA PB = l find the area of triangle OAB where O is the entre of the irle. Ans : ( l x y gx fy ) ( x y gx fy g f l ) ( 53 ) Find the equations of ommon tangents of the two irles ( x ) ( y 3 ) = 4 and x y =. [ Ans: y = 0 3x 4y 5 = 0 x = 0 and 4x 3y 5 = 0 ]