Question

The abscissae and ordinates of the endpoints $A$ and $B$ of a focal chord of the parabola $y^2=4x$ are respectively, the roots of equations $x^2-3x+a=0$ and $y^2+6y+b=0$. The find the equation of the circle with $AB$ as diameter.

Answer 1

$t^2+\frac{1}{t^2}=3$

$2t-\frac{2}{t}=-6$

$t-\frac{1}{t}=3$

$t^2.\frac{1}{t^2}=a$

$a=1$

$2t\left(\frac{-2}{t}\right)=b$

$b=-4$

Equation of circle with $AB$ as diameter $\Rightarrow$

$(x-t^2)(x-1/t^2)+(y-2t)(y+2/t)=0$

$x^2-\frac{x}{t^2}-x{t}^2+1+y^2+\frac{2}{t}y-2ty-4=0$

$x^2+y^2-x(t^2+\frac{1}{t^2})+2y(\frac{1}{t}-t)-3=0$

$x^2+y^2-3x+6y-3=0$