Paragraph: A circle C of radius 1 is inscribed in an equilateral △ PQR . The points of contact of C with the sides PQ , QR , RP are D , E , F respectively. The line PQ is given by the equation 3 x +

Question

Paragraph: A circle C of radius 1 is inscribed in an equilateral △ PQR . The points of contact of C with the sides PQ , QR , RP are D , E , F respectively. The line PQ is given by the equation 3 ​ x + y − 6 = 0 and the point D is ( 2 3 3 ​ ​ , 2 3 ​ ) . Further, it is given that the origin and the centre of C are on the same side of the line PQ . Question: The equation of circle C is

TestHub · Accepted Answer

Let centre of circle C be ( h , k ) . Then, ​ 3 + 1 ​ 3 ​ h + k − 6 ​ ​ = 1 ⇒ 3 ​ h + k − 6 = 2 , − 2 ⇒ 3 ​ h + k = 4 (Rejecting 2 because origin and centre of C are on the same side of PQ ) The point ( 3 ​ , 1 ) satisfies Eq. (i). ∴ Equation of circle C is ( x − 3 ​ ) 2 + ( y − 1 ) 2 = 1 .

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