Question

If all chords of curve ${3x}^2-y^2-2x+4y=0$, which subtend a right angle at the origin, pass through a fixed point. Then the coordinates of the point.

• A. $(\frac{1}{3},2)$
• B. $(1,-2)$
• C. $(2,1)$
• D. $(2,2)$

Answer 1

The correct option is A $(\frac{1}{3},2)$

$3x^2-y^2-2x+4y=0$

$\Rightarrow 3x^2-2x-y^2+4y=0$

$\Rightarrow x^2-\frac{2}{3}x+\frac{1}{9}-\frac{1}{9}-\frac{1}{3}(y^2-4y+4)+\frac{4}{3}=0$

$\Rightarrow {(x-\frac{1}{3})}^2-\frac{1}{3}{(y-2)}^2=\frac{1}{9}-\frac{4}{3}$

$\Rightarrow {(x-\frac{1}{3})}^2-\frac{(y-2)62}{3}=\frac{-11}{9}$

$\Rightarrow \frac{{(x-\frac{1}{3})}^2}{\frac{-11}{9}}-\frac{{(y-2)}^2}{\frac{-11}{3}}=1$

$\therefore$ The required point is $(\frac{1}{3},2)$

Hence, the answer is $(\frac{1}{3},2).$