Mathematics - Parabola Question with Solution | TestHub

Since, equation of normal is $y+xt=2at+a{t}^3$ passes through $(3,0)$. $$\Rightarrow 3t=2t+t^3\Rightarrow t=0,1,-1\text{. }$$ $\therefore P(1,2),Q(0,0),R(1,-2)$. Thus, (i) Area of $△PQR=\frac{1}{2}\times 1\times 4=2$ (ii) Centroid of $△PQR=\left(\frac{2}{3},0\right)$ Equation of circle passing through $P,Q,R$ is $$\begin{array}{rl}& (x-1)(x-1)+(y-2)(y+2)+\lambda (x-1)=0\\ & \Rightarrow 1-4-\lambda =0\Rightarrow \lambda =-3\\ & \therefore \text{ Required equation of circle is }{x}^2+y^2-5x-1=0\\ & \therefore \text{ Centre }\left(\frac{5}{2},0\right)\text{ and radius }\frac{5}{2}.\end{array}$$