Question

The length of the straight line \( x - 3y = 1 \) intercepted by the hyperbola \( x^2 - 4y^2 = 1 \) is

• A. \( \sqrt{10} \)
• B. \( \frac{6}{5} \)
• C. \( \frac{1}{\sqrt{10}} \)
• D. \( \frac{6}{\sqrt{10}} \)

Hint:

To find the length of a line intercepted by a curve, substitute the line's equation into the curve's equation and solve for the points of intersection.

Answer 1

The Correct Option is D

Step 1: Intersection of the line and the hyperbola.
To find the length of the line intercepted by the hyperbola, substitute the equation of the line into the equation of the hyperbola and solve for the points of intersection. The length is then calculated.

Step 2: Conclusion.
Thus, the length of the intercepted line is \( \frac{6}{\sqrt{10}} \). Hence, the correct answer is option (D).

Final Answer: \[ \boxed{\text{(D) } \frac{6}{\sqrt{10}}} \]