Question:
Let $ x + y + 1 = 0 $ and $ x - y + 4 = 0 $ be the asymptotes of a hyperbola $ H $. If $ (1, 1) $ is a point on $ H $, then the length of the latus rectum of $ H $ is
Let $ x + y + 1 = 0 $ and $ x - y + 4 = 0 $ be the asymptotes of a hyperbola $ H $. If $ (1, 1) $ is a point on $ H $, then the length of the latus rectum of $ H $ is
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For hyperbolas, use the equations of the asymptotes to determine the relationship between the constants \( a \) and \( b \), and then calculate the length of the latus rectum using the formula \( \frac{2b^2}{a} \).
Updated On: May 9, 2025
- \( 4\sqrt{3} \)
- \( \sqrt{3} \)
- \( \sqrt{2} \)
- \( \sqrt{5} \)
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The Correct Option is A
Solution and Explanation
The given equation of the hyperbola is formed by the asymptotes: \[ x + y + 1 = 0 \quad \text{and} \quad x - y + 4 = 0 \] These represent the equations of the asymptotes of the hyperbola.
Step 1: The general equation of the hyperbola is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] The asymptotes of the hyperbola are given by the equations: \[ y = \pm \frac{b}{a} x \] From the given equations of the asymptotes, we can determine the relationship between \( a \) and \( b \).
Step 2: Since the length of the latus rectum of a hyperbola is given by \( \frac{2b^2}{a} \), we calculate this using the known values of \( a \) and \( b \) based on the asymptotes.
The final result for the length of the latus rectum is \( 4\sqrt{3} \).
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