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Eq of circle passing through non collinear A,B,C is

Asked by ramasuryakumarivasamsetti | 26 Oct, 2023, 09:19: PM

Expert Answer

Equation of circle :-

begin mathsize 14px style x squared plus y squared plus 2 g space x space plus space 2 f space y space plus space c space equals space 0 end style

. .. . . . . . . . . . . . . . (1)

If the circle given in eqn.(1) is passing through A ( x₁ , y₁) , then we get

begin mathsize 14px style x subscript 1 squared plus y subscript 1 squared plus 2 g space x subscript 1 space plus space 2 f space y subscript 1 space plus space c space equals space 0 end style

............................(2)

Similarly, If the circle given in eqn.(1) is passing through B ( x₂ , y₂) and C ( x₃, y₃) , then we get

begin mathsize 14px style x subscript 2 squared plus y subscript 2 squared plus 2 g space x subscript 2 space plus space 2 f space y subscript 2 space plus space c space equals space 0 end style

.............................. (3)

begin mathsize 14px style x subscript 3 squared plus y subscript 3 squared plus 2 g space x subscript 3 space plus space 2 f space y subscript 3 space plus space c space equals space 0 end style

...............................(4)

We get the following three linear equations from eqn.(2) , eqn.(3) and eqn.(4)

begin mathsize 14px style space 2 g space x subscript 1 space plus space 2 f space y subscript 1 space plus space c space equals space negative x subscript 1 squared minus y subscript 1 squared end style

begin mathsize 14px style space 2 g space x subscript 2 space plus space 2 f space y subscript 2 space plus space c space equals space negative x subscript 2 squared minus y subscript 2 squared end style

begin mathsize 14px style space 2 g space x subscript 3 space plus space 2 f space y subscript 3 space plus space c space equals space negative x subscript 3 squared minus y subscript 3 squared end style

We can solve the above linear equations to get the unknowns g, f, and c in terms of given coordintaes of A, B and C .

By substituting g, f and c in eqn.(1) , we get the required equation of circle that is passing through the points A, B and C

Answered by Thiyagarajan K | 27 Oct, 2023, 07:19: AM

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