03%20-%20Circle

03%20-%20Circle - PROBLEMS (1) 03 - CIRCLE Obtain the...

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PROBLEMS 03 - CIRCLE Page 1 ( 1 ) Obtain the equation of the circle passing through the points ( 5, - 8 ), ( - 2, 9 ) and ( 2, 1 ). [ Ans: x 2 + y 2 + 116x + 48y - 285 = 0 ] ( 2 ) Find the equation of the circumscribed circle of the triangle formed by three lines given by x + y = 6, 2x + y = 4 and x + 2y = 5. [ Ans: x 2 + y 2 - 17x - 19y + 50 = 0 ] ( 3 ) Find centre and radius of the circle whose equation is 4x 2 + 4y 2 - 12x + 24y + 29 = 0. 2 , 3 , 2 3 : Ans - ( 4 ) Find the equation of the circle touching both the axes and passing through ( 1, 2 ). [ Ans: x 2 + y 2 - 2x - 2y + 1 = 0, x 2 + y 2 - 10x - 10y + 25 = 0 ] ( 5 ) The cartesian equation of the circle is x 2 + y 2 + 4x - 2y - 4 = 0. Find its parametric equations. [ Ans: x = - 2 + 3 cos θ , y = 1 + 3 sin θ , θ ( - π, π ] ] ( 6 ) Show that the line-segments, joining any point of a semi-circle to the end points of the diameter, are perpendicular to each other. ( 7 ) Show that the point ( 4, - 5 ) is inside the circle x 2 + y 2 - 4x + 6y - 5 = 0. Find the point on the circle which is at the shortest distance from that point. [ Ans: ( 5, - 6 ) ] ( 8 ) Find the length of the chord of the circle x 2 + y 2 - 4x - 2y - 20 = 0 cut off by the line x + y - 10 = 0. [ Ans: 2 ]
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03 - CIRCLE Page 2 ( 9 ) Find the mid-point of the chord of the circle x 2 + y 2 = 16 cut off by the line 2x + 3y - 13 = 0. [ Ans: ( 2, 3 ) ] ( 10 ) Find the conditions for the line x cos α + y sin α = p to be the tangent to the circle x 2 + y 2 = r 2 . [ Ans: p 2 = r 2 ] ( 11 ) Find the equations of the tangents to the circle x 2 + y 2 = 17 from the point ( 5, 3 ). [ Ans: 4x - y - 17 = 0, x + 4y - 17 = 0 ] ( 12 ) The lengths of the tangents drawn from a point P to two circles with centre at origin are inversely proportional to the corresponding radii. Show that all such points P lie on a circle with centre at origin. ( 13 ) Find the measure of an angle between two tangents to the circle x 2 + y 2 = a 2 drawn from the point ( h, k ). [ Ans: 2 tan - 1 ( a / 2 2 2 a k h - + ) ] ( 14 ) Find the set of all points P outside a circle x 2 + y 2 = a 2 such that the tangents to the circle, drawn from P, are perpendicular to each other. [ Ans: x 2 + y 2 = 2a 2 ] ( 15 ) Find the equation of the circle which passes through the points of intersection of the circles x 2 + y 2 = 13 and x 2 + y 2 + x - y - 14 = 0 and whose centre lies on the line 4x + y - 6 = 0. [ Ans:
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03%20-%20Circle - PROBLEMS (1) 03 - CIRCLE Obtain the...

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