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:
Show Hint
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: Mar 11, 2025
- \( \frac{1}{4} \)
- \( \frac{1}{2} \)
- 4
- 1
- 2
Hide Solution
Verified By Collegedunia
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).
Was this answer helpful?
0
0
Top Questions on Parabola
- Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:
- If the line \( 3x - 2y + 12 = 0 \) intersects the parabola \( 4y = 3x^2 \) at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to:
- Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:
- Let A and B be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^2 = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between A and B, and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^2 = 4x \), then the value of \( (a + d) \) is:
- Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at points A and B, then \( (AB)^2 \) is equal to:
View More Questions
Questions Asked in KEAM exam
- In the measurement of length 6 $\mu$m is equal to $x$ pm. Then the value of $x$ is
- KEAM - 2024
- Dimensional Analysis
- If the lines \( \frac{x-1}{2} = \frac{y-2}{\alpha} = \frac{z-3}{2} \) and \( \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-2} \) are perpendicular, then the value of \( \alpha \) is:
- KEAM - 2024
- Integration
- An organic compound contains 37.5% C, 12.5% H and the rest oxygen. What is the empirical formula of the compound?
- KEAM - 2024
- Oxygen
- Let A, B, C denote the set of students in a college who play football, basketball, and cricket respectively. If \( n(A) = 60 \), \( n(B) = 55 \), \( n(C) = 70 \), \( n(A \cup B \cup C) = 100 \) and \( n(A \cap B \cap C) = 20 \), then the number of students who play exactly two of these sports is
- KEAM - 2024
- Relations and functions
For the reaction:
\[ 2A + B \rightarrow 2C + D \]
The following kinetic data were obtained for three different experiments performed at the same temperature:
\[ \begin{array}{|c|c|c|c|} \hline \text{Experiment} & [A]_0 \, (\text{M}) & [B]_0 \, (\text{M}) & \text{Initial rate} \, (\text{M/s}) \\ \hline I & 0.10 & 0.10 & 0.10 \\ II & 0.20 & 0.10 & 0.40 \\ III & 0.20 & 0.20 & 0.40 \\ \hline \end{array} \]
The total order and order in [B] for the reaction are respectively:
- KEAM - 2024
- Grignard’s reagents and their uses
View More Questions