Question

Normals are drawn at points P, Q and R lying on the parabola $y^2=4x$ which intersect at (3, 0). Then,

Answer 1

Equation of the normal is $y=mx-2m-m^3$, since it passes through $(3,0)$

$3m-2m-m^3=0\Rightarrow m=0,\pm 1$

Then coordinates of $\Delta PQR$ are $P(0,0),Q(1,-2)$ and $R(1,2)$
A) Area $=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ 1& -2& 1\\ 1& 2& 1\end{array}\right|=2$
B) If $(x,y)$ is circumcenter, then

$x^2+y^2={(x-1)}^2+{(y-2)}^2={(x-1)}^2+{(y+2)}^2\Rightarrow x=\frac{5}{2},y=0$

Hence radius is $\frac{5}{2}$
C) Centroid is $(\frac{0+1+1}{3},\frac{0-2+2}{3})=(\frac{2}{3},0)$

Hence distance from vertex is $\frac{2}{3}$
D) Centroid is $(\frac{2}{3},0)$ and Circumcenter is $(\frac{5}{2},0)$

Then distance between centroid and circumcenter is $\frac{11}{6}$