Tiêu đề Circle A & R.pdf ✅

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
Because Statement 2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x2 + y2 = 338. [A] Q.25 Statement I : The number of common tangents to the circles x2 + y2 = 4 and (x –5)2 + y2 = k2 will be 4 only if k<1. because Statement II : Two circles have four common tangents if distance between centres is greater than sum of radii. Sol.[D] Centres are C1(0, 0) & C2(5, 0) Applying d > (r1 + r2)  d = C1C2  5 > k + 2  k < 3 Statement (II) is correct, but statement (I) is false Q.26 Assertion (A) : The number of common tangents to the circle x2 + y2 = 4 and x2 + y2 – 6x – 8y = 24 is 4. Reason (R) : Circles with centre C1, C2 and radii r1, r2 and if |C1C2| > r1 + r2, then circles have 4 common tangents. Sol.[D] C1  (0, 0), r1 = 2  C2 (3, 4), r2 = 7 |C1C2| = 5  r1 + r2 = 9, |r1 – r2| = 5 |C1C2| = |r1 – r2| circle touch internally  Common tangent = 1 Q.27 Statement-I : The equation of chord of the circle x2 + y2 – 6x + 10y – 9 = 0 which is bisected at (–2, 4) must be x + y – 2 = 0. Statement-II : In notation the equation of the chord of the circle S = 0 bisected at (x1, y1) must be T = S1. Sol.[D] x 2 + y2 – 6x + 10y – 9 = 0 & P(–2, 4) equation of chord in mid point form T = S1 – 2x + 4y – 3(x – 2) + 5 (y + 4) – 9= 4 + 16 + 12 + 40 – 9 – 2x + 4y – 3x + 6 + 5y + 20 = 4 + 16 + 12 + 40 – 5x + 9y – 46 = 0 or 5x – 9y + 46 = 0 Statement I is false. Statement II is true. Q.28 Statement 1: (s1) : The circle x 2 + y2 –6x – 4y – 7 = 0 touches y-axis Statement 2: (s2) : The circle x2 + y2 + 6x + 4y – 7 = 0 touches x- axis. Which of the following is a correct statement? (A) Both s1 and s2 are correct (B) Both s1 and s2 are not correct (C) s1 is correct, s2 is wrong (D) s2 is correct, s1 is wrong [B] Q. 29 Statement 1 : (S1) Two points A (10, 0) and origin 'O' are given and a circle is x2 + y2 – 6x + 8y – 11 = 0 then circle always cuts line OA. Statement 2 : (S2) Centre of circle, origin and point A are non collinear. [B] Q. 30 Statement (1): Two points A(10, 0) and O (0, 0) are given and a circle x2 + y2– 6x + 8y – 11= 0. The circle always cuts the line segments OA. Statement (2) : The centre of the circle, point A and the point O are not collinear. [B] Q.31 Statement (1) : If a line L = 0 is a tangents to the circle S = 0 then it will also be a tangent to the circle S + L = 0. Statement (2) : If a line touches a circles then perpendicular distance from centre of the circle on the line must be equal to the radius. [B] Q.32 Consider the following statements:- Statement (1): The circle x2 + y2 = 1 has exactly two tangents parallel to the x-axis Statement (2): dx dy = 0 on the circle exactly at the point (0, ±1). [A] Q.33 Statement (1): The equation of chord of the circle x2 + y2 – 6x + 10y – 9 = 0, which is bisected at (–2, 4) must be x + y – 2= 0. Statement (2) : In notations the equation of the chord of the circle S = 0 bisected at (x1,y1) must be T = S1. [D] Q.34 Statement (1): If two circles x 2 + y2 + 2gx +2fy = 0 and x2 + y2 + 2g'x +2f 'y = 0 touch each other then f 'g = fg'.