Question

The locus of the midpoint of the system of parallel chords parallel to the line \( y = 2x \) to the hyperbola \( 9x^2 - 4y^2 = 36 \) is:

• A. \( 8x - 9y = 0 \)
• B. \( 9x - 8y = 0 \)
• C. \( 8x + 9y = 0 \)
• D. \( 9x - 4y = 0 \)

Hint:

For hyperbolas, use the parametric equations to find midpoints or loci of chords systematically

Answer 1

The Correct Option is B

1. The equation of the hyperbola is:

\[ \frac{x^2}{4} - \frac{y^2}{9} = 1. \]

2. The equation of the chord parallel to \(y = 2x\) is:

\[ y = 2x + c. \]

3. Using the midpoint formula for a hyperbola: - The midpoint satisfies the locus equation derived from substituting \(y = 2x + c\) into the hyperbola equation.

4. After simplification, the locus of the midpoint is:

\[ 9x - 8y = 0. \]