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AP EAMCET201823 Apr 2018Evening ShiftMathematicsCircleActual

x^2+y^2+2 x+4 y-20=0 and x^2+y^2+6 x-8 y+10=0 are the given circles. Which one of the following is correct?

Options

  1. AThey intersect orthogonally and will have two common tangents. The length of their common chord is 5 3 2
  2. BThey intersect at right angles and will have two common tangents. The length of their common chord is 2
  3. CThey do not intersect orthogonally and will have three common tangents. The length of their direct common tang
  4. DThey touch each other internally and will have only one common tangent

Correct answer

A. They intersect orthogonally and will have two common tangents. The length of their common chord is 5 3 2

Step-by-step solution

The equation of given two circles are x^2+y^2+2 x+4 y-20=0 and x^2+y^2+6 x-8 y+10=0 aligned 2 g₁ g₂+2 f₁ f₂ & =6-16 & =-10=c₁+c₂ aligned So, circles intersects each other orthogonally and will have two common tangents. Now, equation of common chord is 2 x-6 y+15=0 So, length of common chord is aligned & 2 25- (-2+12+15)^2 4+36 = & 2 25- 625 40 =2 1000-625 40 =5 3 2 . aligned

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