To solve this problem, we need to determine the fixed point through which all chords of the given hyperbola, subtending a right angle at the origin, pass.
The equation of the hyperbola provided is:
\(3x^2 - 2y^2 - 4x + y = 0\)
First, let's rearrange this equation into a standard form:
- Complete the square for the \(x\) terms:
- \(3x^2 - 4x = 3(x^2 - \frac{4}{3}x)\)
- Complete the square: \(3(x - \frac{2}{3})^2 - \frac{4}{3}\)
- The equation becomes: \(3(x - \frac{2}{3})^2 - \frac{4}{3} - 2y^2 + y = 0\)
- Rearranging, \(3(x - \frac{2}{3})^2 - 2(y^2 - \frac{y}{2}) - \frac{4}{3} = 0\)
- Complete the square for \(y\):
- \(-2(y^2 - \frac{y}{2}) = -2((y - \frac{1}{4})^2 - \frac{1}{16})\)
- The equation becomes: \(3(x - \frac{2}{3})^2 - 2(y - \frac{1}{4})^2 = \frac{4}{3} + \frac{2}{16} = \frac{11}{12}\)
The standard form of the hyperbola is thus established, centered at \((\frac{2}{3}, \frac{1}{4})\).
Next, consider a chord of the hyperbola that subtends a right angle at the origin:
- For a chord \(PQ\) where \(P(x_1, y_1)\) and \(Q(x_2, y_2)\), the condition for a right angle at the origin is: \(x_1 \cdot x_2 + y_1 \cdot y_2 = 0\)
Derive the equation of the chord:
- Equation of chord with a midpoint at \((h, k)\) is: \(T = S_1\)
- This transforms in hyperbola equation: \(3hx - 2ky - 4x + y = 3h^2 - 2k^2 - 4h + k\)
To find the fixed point, solve with the condition \(x_1x_2 + y_1y_2 = 0\):
- Calculate coordinates that solve this independent of anyone chord.
After solving these conditions and equations simultaneously, we find that, in this instance, the intervals obtained for fixed points do not match any of the given options, concluding with:
Correct Answer: None of the above