Suppose AB is a focal chord of the parabola \( y^2 = 12x \) of length \( l \) and slope \( m<\sqrt{3} \). If the distance of the chord AB from the origin is \( d \), then \( ld^2 \) is equal to _________.
Correct Answer: 108
Solution and Explanation
For a focal chord of the parabola \(y^2 = 12x\), we have:
\[ l = 4a \csc^2 \theta \]
Given that \(l = 12x\) and using the property of a focal chord, we find:
\[ l = 12 \times \frac{9}{d^2} \]
Thus:
\[ l d^2 = 108 \]
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