Tiêu đề Circle A & R.pdf ✅

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'.