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:
Show Hint
- \( 8x - 9y = 0 \)
- \( 9x - 8y = 0 \)
- \( 8x + 9y = 0 \)
- \( 9x - 4y = 0 \)
The Correct Option is B
Solution and Explanation
1. The hyperbola's equation:
\[ \frac{x^2}{4} - \frac{y^2}{9} = 1. \]
2. The chord's equation, parallel to \(y = 2x\):
\[ y = 2x + c. \]
3. Midpoint analysis: The midpoint adheres to the locus equation, derived by inserting \(y = 2x + c\) into the hyperbola's equation.
4. The simplified midpoint locus is:
\[ 9x - 8y = 0. \]
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