For each parabola y=x^2+p x+q , meeting coordinate axes at 3-distinct points, if circles are drawn through these points, then the family of circles must pass through

Step-by-step solution

Suppose the parabola y=x^2+p x+q cuts X -axis at A( , 0) and B( , 0) . Then, , are roots of the equation x^2+p x+q=0 + =-p and =q The parabola y=x^2+p x+q cuts Y -axis at (0, q) . Let the equation of the circle passing through A, B and C be aligned & x^2+y^2+2 g x+2 f y+c =0 &...(i) & ^2+2 g +c =0 &...(ii) & ^2+2 g +c =0 &...(iii) and & q^2+2 f q+c =0 &...(iv) aligned Subtracting Eq. (iii) from Eq. (ii), we get array rlrl + +2 g & =0 & g = p / 2 array Adding Eqs. (ii) and (iii), we get array r ^2+ ^2+2 g( + )+2 c=0