All chords of the curve $3x^2-y^2-2x+4y=0$ which subtend a right angle at the origin pass through a fixed point $P(\alpha ,\beta ).$ Which of the following statement(s) is (are) CORRECT?

Consider the chord as $y=mx+c$ intersects the curve $3x^2-y^2-2x+4y=0$ at A and B.

$\overline{)AOB}=\frac{\pi }{2}$

Homogenize the curve with the chord to get pair of lines OA and OB

$3x^2-y^2-2x\left(1\right)+4y\left(1\right)=0$

$3x^2-y^2-2x\left(\frac{y-mx}{c}\right)+4y\left(\frac{y-mx}{c}\right)=0\Rightarrow x^2(3+\frac{2m}{c})+y^2(-1+\frac{4}{c})+xy(\frac{-2}{c}-\frac{4m}{c})=0$$\overline{)AOB}=\pi /2$

coefficient of $x^2$ + coefficient of $y^2=0$

$3+\frac{2m}{c}+(-1+\frac{4}{c})=0\Rightarrow 2+\frac{2m}{c}+\frac{4}{c}=0\Rightarrow c+m+2=0$

$y=mx+c\Rightarrow mx-y+c=0$

The chords pass through P (1,-2)

A) $x^2+y^2-2x+4y-4=0$

Q, R lies on circle
P is centre of circle
PQ + PR = r + r = 2r = 2 (3) = 6
B) Maximum area of $\Delta \mathcal{l}e PQR=\left(\frac{1}{2}\right)\left(3\right)\left(3\right)\sin \theta =\frac{9}{2}\sin \theta \le \frac{9}{2}@\theta =\frac{\pi }{2}$

C) $x^2+y^2=5$

$P(1,-2)$ lies on the circle

$PQ+PR\in (0,4\sqrt{5})$

D) Maximum area of $\Delta \mathcal{l}e$ PQR = equilateral triangle PQR

$=\left(\frac{\sqrt{3}}{4}\right){\left(\sqrt{3}r\right)}^2=\left(\frac{\sqrt{3}}{4}\right)\left(3r^2\right)=\frac{\left(3\right)\left(5\right)\sqrt{3}}{4}=\frac{15\sqrt{3}}{4}$ sq. units