Program to find the length of Latus Rectum of a Parabola

Given a, b, and c the coefficients of X², X, and the constant term in the general equation of the parabola y = a\times x^{2} + b\times x + c , the task is to find the length of the Latus Rectum of the parabola.

Examples:

Input: a = 3, b = 5, c = 1
Output: 0.333333

Input: a = 4, b = 0, c = 4
Output: 0.25

Approach: The given problem can be solved based on the following observations:

Observation:

The Latus rectum of a parabola is the perpendicular line to the axis and at the focus of parabola and its length is equal to 4 times the distance between the focus and vertex of the parabola.

Therefore, the task is reduced to find the distance between focus and vertex of the parabola using formula:
d = sqrt( (x_{1}-x_{2})^{2} + (y_{1}-y_{2})^{2} )

Follow the steps below to solve the problem:

Initialize two variables, say vertex and focus to store the coordinates of vertex and focus of the parabola.
Find the coordinates of the vertex and focus of the parabola and store in corresponding variables.
Initialize a variable, say length, and set it to 4 times the distance between vertex and focus of the parabola.
Print the value of length as the answer.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

#include <bits/stdc++.h>
using namespace std;

// Function to calculate distance
// between two points
float distance(float x1, float y1,
               float x2, float y2)
{
    // Calculating distance
    return sqrt((x2 - x1) * (x2 - x1)
                + (y2 - y1) * (y2 - y1));
}

// Function to calculate length of
// the latus rectum of a parabola
float lengthOfLatusRectum(float a,
                          float b, float c)
{
    // Stores the co-ordinates of
    // the vertex of the parabola
    pair<float, float> vertex
        = { (-b / (2 * a)),
            (((4 * a * c) - (b * b)) / (4 * a)) };

    // Stores the co-ordinates of
    // the focus of parabola
    pair<float, float> focus
        = { (-b / (2 * a)),
            (((4 * a * c) - (b * b) + 1) / (4 * a)) };

    // Print the distance between focus and vertex
    cout << 4 * distance(
                    focus.first, focus.second,
                    vertex.first, vertex.second);
}

// Driver Code
int main()
{
    // Given a, b & c
    float a = 3, b = 5, c = 1;

    // Function call
    lengthOfLatusRectum(a, b, c);

    return 0;
}

Java

// Java program for the above approach
class GFG{
  static class pair
  { 
    float first;
    float second; 
    public pair(float first, float second)  
    { 
      this.first = first; 
      this.second = second; 
    }    
  } 

  // Function to calculate distance
  // between two points
  static float distance(float x1, float y1,
                        float x2, float y2)
  {

    // Calculating distance
    return (float) Math.sqrt((x2 - x1) * (x2 - x1)
                             + (y2 - y1) * (y2 - y1));
  }

  // Function to calculate length of
  // the latus rectum of a parabola
  static void lengthOfLatusRectum(float a,
                                  float b, float c)
  {

    // Stores the co-ordinates of
    // the vertex of the parabola
    pair vertex
      = new pair( (-b / (2 * a)),
                 (((4 * a * c) - (b * b)) / (4 * a)) );

    // Stores the co-ordinates of
    // the focus of parabola
    pair focus
      = new pair( (-b / (2 * a)),
                 (((4 * a * c) - (b * b) + 1) / (4 * a)) );

    // Print the distance between focus and vertex
    System.out.print(4 * distance(
      (float)focus.first, (float)focus.second,
      (float)vertex.first, (float)vertex.second));
  }

  // Driver Code
  public static void main(String[] args)
  {

    // Given a, b & c
    float a = 3, b = 5, c = 1;

    // Function call
    lengthOfLatusRectum(a, b, c);

  }
}

// This code is contributed by 29AjayKumar

Python3

# Python 3 program for the above approach
from math import sqrt

# Function to calculate distance
# between two points
def distance(x1, y1, x2, y2):
  
    # Calculating distance
    return sqrt((x2 - x1) * (x2 - x1) + (y2 - y1) * (y2 - y1))

# Function to calculate length of
# the latus rectum of a parabola
def lengthOfLatusRectum(a, b, c):
  
    # Stores the co-ordinates of
    # the vertex of the parabola
    vertex =  [(-b / (2 * a)), (((4 * a * c) - (b * b)) / (4 * a))]

    # Stores the co-ordinates of
    # the focus of parabola
    focus = [(-b / (2 * a)), (((4 * a * c) - (b * b) + 1) / (4 * a))]

    # Print the distance between focus and vertex
    print("{:.6f}".format(4 * distance(focus[0], focus[1], vertex[0], vertex[1])))

# Driver Code
if __name__  == "__main__":
  
    # Given a, b & c
    a = 3
    b = 5
    c = 1

    # Function call
    lengthOfLatusRectum(a, b, c)
    
    # This code is contributed by bgangwar59.

// C# program for the above approach
using System;

public class GFG{
  class pair
  { 
    public float first;
    public float second; 
    public pair(float first, float second)  
    { 
      this.first = first; 
      this.second = second; 
    }    
  } 

  // Function to calculate distance
  // between two points
  static float distance(float x1, float y1,
                        float x2, float y2)
  {

    // Calculating distance
    return (float) Math.Sqrt((x2 - x1) * (x2 - x1)
                             + (y2 - y1) * (y2 - y1));
  }

  // Function to calculate length of
  // the latus rectum of a parabola
  static void lengthOfLatusRectum(float a,
                                  float b, float c)
  {

    // Stores the co-ordinates of
    // the vertex of the parabola
    pair vertex
      = new pair( (-b / (2 * a)),
                 (((4 * a * c) - (b * b)) / (4 * a)) );

    // Stores the co-ordinates of
    // the focus of parabola
    pair focus
      = new pair( (-b / (2 * a)),
                 (((4 * a * c) - (b * b) + 1) / (4 * a)) );

    // Print the distance between focus and vertex
    Console.Write(4 * distance(
      (float)focus.first, (float)focus.second,
      (float)vertex.first, (float)vertex.second));
  }

  // Driver Code
  public static void Main(String[] args)
  {

    // Given a, b & c
    float a = 3, b = 5, c = 1;

    // Function call
    lengthOfLatusRectum(a, b, c);

  }
}

 

// This code is contributed by 29AjayKumar

JavaScript

<script>
    // Javascript program for the above approach
    
    // Function to calculate distance
    // between two points
    function distance(x1, y1, x2, y2)
    {
        // Calculating distance
        return Math.sqrt((x2 - x1) * (x2 - x1)
                    + (y2 - y1) * (y2 - y1));
    }

    // Function to calculate length of
    // the latus rectum of a parabola
    function lengthOfLatusRectum(a, b, c)
    {
        // Stores the co-ordinates of
        // the vertex of the parabola
        let vertex
            = [ (-b / (2 * a)),
                (((4 * a * c) - (b * b)) / (4 * a)) ];

        // Stores the co-ordinates of
        // the focus of parabola
        let focus
            = [ (-b / (2 * a)),
                (((4 * a * c) - (b * b) + 1) / (4 * a)) ];

        // Print the distance between focus and vertex
        document.write((4 * distance(
                        focus[0], focus[1],
                        vertex[0], vertex[1])).toFixed(6));
    }
    
    // Given a, b & c
    let a = 3, b = 5, c = 1;
 
    // Function call
    lengthOfLatusRectum(a, b, c);
 
 // This code is contributed by divyeshrabadiya07.
</script>

Output:

0.333333

Time Complexity: O(logn) because it is using inbuilt sqrt function
Auxiliary Space: O(1)

Program to find the length of Latus Rectum of a Parabola

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