Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main202331 Jan 2023Morning ShiftMathematicsCircleActual

Let a circle C 1 be obtained on rolling the circle x 2 + y 2 - 4 x - 6 y + 11 = 0 upwards 4 units on the tangent T to it at the point 3 , 2 . Let C 2 be the image of C 1 in T . Let A and B be the centers of circles C 1 and C 2 respectively, and M and N be respectively the feet of perpendiculars drawn from A and B on the x -axis. Then the area of the trapezium AMNB is:

Options

  1. A2 2 + 2
  2. B4 1 + 2
  3. C3 + 2 2
  4. D2 1 + 2

Correct answer

B. 4 1 + 2

Step-by-step solution

We have, x 2 + y 2 - 4 x - 6 y + 11 = 0 ⇒ x - 2 2 + y - 3 2 = 2 Centre ≡ 2 , 3 Radius r = 2 units Equation of tangent at 3 , 2 is 3 x + 2 y - 2 x + 3 - 3 y + 2 + 11 = 0 ⇒ x = y = 1       . . . 1 On rolling the given circle upward 4 units on the tangent 1 , centre of the circle also moving 4 unit upward, so Centre of C 1 is = 2 + 2 2 , 3 + 2 2 A 2 + 2 2 , 3 + 2 2 Centre of C 2 is B 4 + 2 2 , 1 + 2 2 So, M 2 + 2 2 , 0 and N 4 + 2 2 , 0 . ∴ area of trapezium = 1 2 A M + B N &#215

Practice Circle on Quantrex Academy →

More from Circle

Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) The circle with centre (1, 2) and touching the straight line 3x + 4y = 2026Consider the circle C: x^2+y^2-6x-8y-11=0 . Let a variable chord AB of the circle C subtend a right angle at the origin. If the locus of the foot of the perpendicular drawn from th 2026Let C be a circle having centre in the first quadrant and touching the x -axis at a distance of 3 units from the origin. If the circle C has an intercept of length 6 3 on y -axis, 2026Let the line x - y = 4 intersect the circle C: (x-4)^2 + (y+3)^2 = 9 at the points Q and R . If P( , ) is a point on C such that PQ = PR , then (6 + 8 )^2 is equal to __________. 2026Let the centre of the circle x^2 + y^2 + 2gx + 2fy + 25 = 0 be in the first quadrant and lie on the line 2x - y = 4 . Let the area of an equilateral triangle inscribed in the circl 2026Let P be a moving point on the circle x^2 + y^2 - 6x - 8y + 21 = 0 . Then, the maximum distance of P from the vertex of the parabola x^2 + 6x + y + 13 = 0 is equal to: 2026Suppose that two chords, drawn from the point (1, 2) on the circle x^2 + y^2 + x - 3y = 0 are bisected by the y -axis. If the other ends of these chords are R and S , and the mid p 2026Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines x + (k-1)y + 3 = 0 and 2x + k^2 y - 4 = 0 . If the line x - y + 2026 Full Circle list All JEE Main PYQs