Question

All chords of the curve ${3x}^2$-$y^2$-2x+4y = 0 that subtends a right angle at the origin, pass through a fixed point whose coordinates are

• A. (1, −2)
• B. (1, 2)
• C. (-1, 2)
• D. (-1,- 2)

Answer 1

The correct option is A (1, −2)

Let the equation of chord by y = mx + c. Combined equation of line joining the point intersection with origin is
$3x^2-y^2-2(x-2y)\left(\frac{y-mx}{c}\right)=0i.e.,x^2(3c+2m)-y^2(c-4)-2xy(1+2m)=0$
These lines will be mutually perpendicular, if
3c + 2m – c + 4 = 0
$\Rightarrow$ 2m + 2c = −4
$\Rightarrow$ m + c = −2
That means the chord y = mx + c is always pass through the point (1, −2).
Hence, (a) is the correct answer.