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Most Important Selected Qs for JEE AdvancedMathematicsCircle

Consider the circles S₁: x^2+y^2=4 and S₂: x^2+y^2-2 x-4 y+4=0 which of the following statements are correct?

Options

  1. ANumber of common tangents to these circles is 2.
  2. BIf the power of a variable point P w.r.t. these two circles is same then P moves on the line x+2 y-4=0
  3. CSum of the y -intercepts of both the circles is 6.
  4. DThe circles S ₁ and S ₂ are orthogonal.

Correct answer

D. The circles S ₁ and S ₂ are orthogonal.

Step-by-step solution

aligned & S₁: x^2+y^2=4 and S₂=x^2+y^2-2 x-4 y+4=0 & centre: (0,0) ; radius =2 centre :(1,2) ; radius =1 aligned (A) d = distance between centres = 5 r₁+r₂=3 ₁+r₂=3 |r₁-r₂ |=1 |r₁-r₂ | d r₁+r₂ these 2 circles are intersecting. number of common tangents is 2 . (B) P ( h , k ) power of point P is same w.r.t. these two circles. h^2+k^2-4 = h^2+k^2-2 h-4 k+4 -4=-2 h-4 k+4 aligned & 2 h+4 k-8=0 & x+2 y-4=0 aligned (C) y intercept of S₁ is 2 +4 =4 intercept of S ₂ is 2 4-4 =0 sum of y -intercept =4 (D) 2(0+0)=-4+4 circle

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