How to find the latus rectum of a parabola?
Understand the Problem
The question is asking how to calculate the latus rectum of a parabola, which is a measure of the width of the parabola at its focus. We can find this by using the formula related to the definition of the parabola and its focus.
Answer
The length of the latus rectum is $L = \frac{1}{|a|}$.
Answer for screen readers
The length of the latus rectum is given by:
$$ L = \frac{1}{|a|} $$
Steps to Solve
- Identify the standard form of the parabola
The standard form for a parabola that opens upwards or downwards is given by the equation:
$$ y = ax^2 $$
- Determine the value of (a)
From the equation of the parabola, the value of (a) can often be identified directly. If the equation is given in a different form, convert it to the standard form to identify (a).
- Use the formula for the latus rectum
The length of the latus rectum (L) can be calculated using the formula:
$$ L = \frac{1}{|a|} $$
- Substitute the value of (a)
After determining (a) from the standard form of the parabola, substitute this value into the latus rectum formula to compute (L).
- Calculate the final length
Perform the calculation to find the length of the latus rectum based on the substituted value.
The length of the latus rectum is given by:
$$ L = \frac{1}{|a|} $$
More Information
The latus rectum is significant in the study of conics, particularly in understanding the geometry of parabolas. It offers insights into the distance and relationship between the focus and the directrix of the parabola.
Tips
- Failing to correctly identify the value of (a) from the parabola's equation.
- Miscalculating the absolute value when substituting into the latus rectum formula.
