Question:
The length of the latus rectum of the parabola \( y^2 = x \) is:
The length of the latus rectum of the parabola \( y^2 = x \) is:
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For parabolas of the form \( y^2 = 4ax \), the length of the latus rectum is given by \( 4a \). Compare the given equation with the standard form to find the value of \( a \) and then compute the length of the latus rectum.
Updated On: Apr 16, 2025
- \( \frac{1}{4} \)
- \( \frac{1}{2} \)
- 4
- 1
- 2
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The Correct Option is D
Solution and Explanation
The equation of the given parabola is:
\[
y^2 = x.
\]
This is a standard parabola of the form \( y^2 = 4ax \), where the vertex is at the origin \( (0, 0) \) and the focus is at \( (a, 0) \).
Step 1: Comparing the equation \( y^2 = x \) with \( y^2 = 4ax \), we find that \( 4a = 1 \), so: \[ a = \frac{1}{4}. \]
Step 2: The length of the latus rectum of a parabola is given by the formula \( 4a \). Therefore, the length of the latus rectum is: \[ 4a = 4 \times \frac{1}{4} = 1. \]
Thus, the length of the latus rectum is 1.
Therefore, the correct answer is option (D).
This is a standard parabola of the form \( y^2 = 4ax \), where the vertex is at the origin \( (0, 0) \) and the focus is at \( (a, 0) \).
Step 1: Comparing the equation \( y^2 = x \) with \( y^2 = 4ax \), we find that \( 4a = 1 \), so: \[ a = \frac{1}{4}. \]
Step 2: The length of the latus rectum of a parabola is given by the formula \( 4a \). Therefore, the length of the latus rectum is: \[ 4a = 4 \times \frac{1}{4} = 1. \]
Thus, the length of the latus rectum is 1.
Therefore, the correct answer is option (D).
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