Tiêu đề Circle A & R.pdf ✅

MATHEMATICS The following questions given below consist of an "Assertion" (A) and "Reason" (R) Type questions. Use the following Key to choose the appropriate answer. (A) If both (A) and (R) are true, and (R) is the correct explanation of (A). (B) If both (A) and (R) are true but (R) is not the correct explanation of (A). (C) If (A) is true but (R) is false. (D) If (A) is false but (R) is true. Q.1 Assertion : Let A, B, C be three points on a line, B lying between A and C. Consider all circles passing through B and C, the points of contact P of the tangents from A to circle lie on a circle. Reason : Because the line segment joining centre of circle and A point subtends 90o at point of contact P. Q.2 Assertion : Circles S1 : x2 + y2 = 4 and S2 : x 2 + y2 – 6x – 8y – 24 = 0 have only one common tangent. Reason : Circles have only one tangent if they touch each other internally. [A] Q.3 Assertion : For the circle x2 + y2 – 2x – 6y + 1 = 0, the chord of minimum length and passing through (1, 2) is of length 4 2 . Reason : Any chord passing through (1, 2) of minimum length also passes through the centre of circle. [C] Q.4 Assertion : If the circles ax2 + ay 2 + 2bx + 2cy = 0 and Ax2 + Ay2 + 2Bx + 2Cy = 0 touch each other then cB = bC. Reason : If two circles touch to each other then slope of tangents at their point of contact will be equal for both circles. [A] Q.5 Let C1 be the circle with centre O1 (0, 0) and radius 1 and C2 be the circle with centre O2 (t, t2 + 1) (t  R) and radius 2. Assertion : Circles C1 and C2 always have at least one common tangent for any value of t. Reason : For the two circles O1O2  | r1 – r2 | where r1 and r2 are their radii for any value of t. [A] Sol. Here (O1O2) 2 = t2 + (t2 + 1)2 = t4 + 3t2 + 1  0  O1O2  1 and | r1 – r2 | = 1  O1O2  | r1 – r2 | Thus, the two circles have at least one common tangent. Hence, code (A) is the correct answer. Q.6 Consider a circle S : (x – 2)2 + (y – 3)2 = 13 and a line L : y = x – 12. Assertion : Chord of contact of pair of tangents drawn from every point on L = 0 to S = 0 passes through P(3, 2) Reason : Pole of polar L = 0 with respect to S = 0 is P(3, 2) [A] Sol. x 2 + y2 – 4x – 6y = 0 Let pole of L = 0 is (x1, y1) So its equation xx1 + yy1 – 2(x + x1) – 3(y + y1) = 0 x(x1 – 2) + y(y1 – 3) – (2x1 + 3y1) = 0....(i) x – y – 12 = 0 ....(ii) (i) and (ii) represents the same line so 1 x1 − 2 = 1 y1 3 − − = 12 2x1 + 3y1  x1 = 3, y1 = 2 So pole is (3, 2) If polar of P(3, 2) passes through Q lying on L = 0, then polar of Q will pass through P, so Assertion is also true and Reason is its correct explanation. Q.7 Assertion : If S1 and S2 are non-concentric circles then their radical axis must exist. Reason : S1,S2, S3 are three circles such that no two are concentric then their radical centre is defined. [C] Sol. S1, S2, S3 have centres on a line then radical centre does not exist.
Q.8 Assertion : Suppose ABCD is a cyclic quadrilateral inscribed in a circle of radius one unit with AB.CD+ BC.DA  4, then ABCD is a square. Reason : A cyclic quadrilateral is a square if its diagonals are the diameters of the circle. Sol. [C] Since ac + bd = AC. BD  4 but ac + bd  4 (AM  GM)  AC = BD = 2 and ac = bd = 2 Q.9 Assertion : Two distinct chords drawn from the point (3, 1) on the circle x 2 + y2 – 3x – y = 0 are bisected by the x-axis. Reason : If point of bisection is (h, 0) then equation of chord given by T = S1 passing through (3, 1) will be quadratic in h giving two distinct values of R. [A] Q.10 Assertion : Angle between line x + y = 3 and circle x2 + y2 – 2x – 4y – c 2 = 0 will not depend on c - Reason : As line passes through centre of circle so angle is 90o [A] Q.11 Assertion : The number of common tangents to the circles x2 + y2 = 4 and x2 + y2 – 6x – 8y = 24 is 1. Reason : If two circles touch other externally then of common tangents 3. [D] Q.12 Assertion : If three circles which are such that their centres are non-colliner, then exactly one circle exists which cuts the three circles orthogonally. Reason : Radical axis for two intersecting circles is the common chord. [B] Q.13 Assertion : If a line L = 0 is tangent to the circle S = 0, then it will also be a tangent to the circle S + L = 0. Reason : If a line touches a circle, then perpendicular distance of the line from the centre of the circle is equal to the radius of the circle. [A] Q.14 Tangents are drawn from the point (17,7) to the circle x2 + y2 = 169. [IIT 2007] Assertion : The tangents are mutually perpendicular. Reason : The locus of the points from which mutually perpendicular tangents can be drawn to given circle is x2 + y2 = 338. [B] Q.15 Consider L1 : 2x + 3y + p – 3 = 0 L2 : 2x + 3y + p + 3 = 0 where p is a real number, and C : x2 + y2 + 6x – 10y + 30 = 0. Assertion : If line L1 is a chord of circle C, then line L2 is not always a diameter of circle C. Reason : If line L1 is a diameter of circle C, then line L2 is not a chord of circle C. [IIT-2008] [C] Q.16 Assertion : The farthest point on the circle x 2 + y2 – 2x – 4y + 4 = 0 from the origin is         + + 5 1 , 2 5 1 1 Reason : The top most vertical point on a circle is the point situated at maximum distance from origin Sol.[C] Reason is false. points lying at minimum and maximum distance from origin lie along the line joining origin and centre Assertion is true. From Reason point is (x, y) r 2 / 5 y 2 1/ 5 x 1 = − = − put r =1 to get the point as         + + 5 1 , 2 5 1 1 Q.17 Assertion (A) : The angle between the tangents drawn from a point (1, 1) lying on the director circle to the circle is /2.
Reason (R) : Angle between the tangents drawn from any point lying in the region between a circle and its director circle is given by tan–1         − 2 2 L R 2RL , where R is radius of circle and L is length of tangent drawn. Sol.[C] (A) is true as (1, 1) lies on the director circle. (R) is false because angle between two tangents is obtuse. Q.18 Assertion(A): If x2 + y2 + 2gx + 2fy = 0 and x 2 + y2 + 2gx + 2f y = 0 touch each other, then f g = fg. Reason(R): Two circles touch each other if line joining their centres is perpendicular to some common tangent Sol. [A] Common point of these circles is (0, 0). Common tangent at (0, 0) is gx + fy = 0 it's slope is m1 =       − f g slope of line joining their centers is −  −  = g g f f m2 here m1m2 = – 1 Q.19 Assertion (A) : Two tangents are drawn from a point on the circle x2 + y2 + 2x + 2y – 6 = 0 to the circle x2 + y2 + 2x + 2y – 2 = 0 then tangents are always perpendicular. Reason (R) : I st circle is the director circle of IInd circle. Sol. [A] Q.20 Assertion (A) : The equation of three circles are x 2 + y2 – 12x – 16y + 64 = 0, 3x2 + 3y2 – 36x + 81 = 0 and x2 + y2 – 16x + 81 = 0. The coordinates of the point from which the length of tangent drawn to each of the three circles are equal, is (33/4, 2). Reason (R) : Radical centre is the point from where tangents drawn to three circles are of equal length. Sol.[D] Q.21 Assertion (A) : The circle of smallest radius passing through two given point A and B must be of radius 2 1 AB. Reason (R) : A straight line is a shortest distance between two points. Sol.[B] Easy Q.22 Assertion : Limiting points of a family of co- axial system of circles are (1, 1) and (3, 3). The member of this family passing through the origin is 2x 2 + 2y2 – 3x – 3y = 0. Reason : Limiting points of a family of coaxial circles are the centres of the circles with zero radius. Sol.[A] Member of family are S1  (x – 1)2 + (y – 1)2 = 0 S2 = (x – 3)2 + (y – 3)2 = 0 circle is S1 + S2 = 0 through (0, 0) Q.23 Statement-1 (S1) : If incircle of ABC touches sides BC, CA & AB at points P, Q & R such that BP = 4, CQ = 5, AR = 6 then its perimeter is 30. Statement-2 (S2) : Length of tangents drawn from external point are same. (A) Both S1 and S2 are correct and S2 is correct explanation of the S1. (B) Both S1 and S2 are correct and S2 is not correct explanation of the S1. (C) S1 is correct but S2 is wrong. (D) S1 is wrong but S2 is correct. Sol.[A] Sides are (4 + 5), (5 + 6) & (6 + 4)  perimeter = 9 + 11 + 10 = 30 Q.24 Tangents are drawn from the point (17, 7) to the circle x2 + y2 = 169. Statement 1 : The tangents are mutually perpendicular