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If circle S 1 has radius R 1 and circle S 2 has radius R 2 then prove that circles [S 1 /R 1 + S 2 /R 2 ]and [S 1 /R 1 - S 2 /R 2 ] are orthogonal

If circle S1 has radius R1 and circle S2 has radius R2 then prove that circles

[S1/R1 + S2/R2]and [S1/R1 - S2/R2] are orthogonal


1 Answers

Vijay Luxmi Askiitiansexpert
357 Points
12 years ago

Dear Student,


        Let us take a general case of a circle.


        Say the equation of the circle s, is


                x2 + y2 + 2gx + 2fy + c = 0


                where radius R1 =  (g2 + f2 - c)1/2


        and say S2 is


                x2 + y2 + 2g1x + 2f1y + c1 = 0


        where radius R2 = (g12 + f12 - c1)1/2


        Now in order to prove that the curves (S1/R1 + S2/R2 )


and (S1/R1 + S2/R2 ) cut each other orthogonally we have to show that dy/dx  to the first curve at the point of intersection and dy/dx to the second curve at the same point when multiplied = –1


        i.e., (dy/dx)c1 * (dy/dx)c2 = –1


Note : Students are advised to remember this property as a result.


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