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 specific geometric parameter. To solve this, we will need to use the standard formulas relating to parabolas.
Answer
For a vertical parabola, the latus rectum \(L = \frac{1}{|a|}\); for a horizontal parabola, \(L = |4a|\).
Answer for screen readers
The latus rectum (L) can be calculated as follows, depending on the orientation of the parabola:
- For a vertical parabola, (L = \frac{1}{|a|})
- For a horizontal parabola, (L = |4a|)
Steps to Solve
- Identify the standard form of the parabola
To find the latus rectum, we first need to know the standard equation of the parabola. The two common forms are:
- For a vertical parabola (opens up or down): $y = ax^2$
- For a horizontal parabola (opens left or right): $x = ay^2$
- Define the value of (a)
Once we have the standard form, we can identify the value of (a). This parameter determines how wide or narrow the parabola is.
- Use the formula for latus rectum
The latus rectum of a parabola can be calculated using the formula:
- For a vertical parabola: $L = \frac{1}{|a|}$
- For a horizontal parabola: $L = |4a|$
- Substitute (a) into the formula
After determining the value of (a), we'll substitute it into the appropriate formula to get the latus rectum.
- Final calculation
Carry out the arithmetic calculations to find the final value of the latus rectum.
The latus rectum (L) can be calculated as follows, depending on the orientation of the parabola:
- For a vertical parabola, (L = \frac{1}{|a|})
- For a horizontal parabola, (L = |4a|)
More Information
The latus rectum is useful in understanding the geometric properties of parabolas, such as their width and focus. It represents the distance across the parabola at the focus.
Tips
- Forgetting to take the absolute value of (a) in the vertical parabola formula can lead to incorrect results.
- Confusing the formulas for vertical and horizontal parabolas, which may result in using the wrong calculations.
