BITSAT Mathematics Question (2022) — Solution

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 \\ ( + )^2-2 +2 g( + )+2 c=0 \\ p^2-2 q-p^2+2 c=0 array [ + =p and g= p 2 ] Putting c=q in Eq. (iv), we get f=- ( q+1 2 ) Substituting the values of g, f and c in Eq. (i), we obtain the equation of family of circles passing through A, B and C as x^2+y^2+p x-(q+1) y+q=0 Clearly, it passes through (0,1).