Line Passing Through 2 Points

Given two points P(x1, y1) and Q(x2, y2) in the coordinate plane, find the equation of the line passing through both the points. 

Examples: 

Input: x1 = 3, y1 = 2, x2 = 2, y2 = 6
Output: 4x+1y = 14
Explanation: The unique line passing through the points (3,2) and (2,6) is 4x+1y=14.

Input: x1 = 3, y1 = 2, x2 = 5, y2 = 7
Output: 5x-2y = 11
Explanation: The unique line passing through the points (3,2) and (5,7) is 5x-2y=11.

Let the given two points be P(x₁, y₁) and Q(x₂, y₂). Now, we find the equation of line formed by these points.   

Let the two points satisfy the line ax + by = c. So, we have, 
ax₁ + by₁ = c 
ax₂ + by₂ = c 

We can set the following values so that all the equations hold true,  a = y₂ - y₁
b = x₁ - x₂
c = ax₁ + by₁

How does this work?

These can be derived by first getting the slope directly and then finding the intercept of the line. OR these can also be derived cleverly by a simple observation as under: 

ax₁ + by₁ = c ...(i)
ax₂ + by₂ = c ...(ii)
Equating (i) and (ii),
ax₁ + by₁ = ax₂ + by₂
=> a(x₁ - x₂) = b(y₂ - y₁)
Thus, for equating LHS and RHS, we can simply have,
a = (y₂ - y₁)
AND
b = (x₁ - x₂)
so that we have,
(y₂ - y₁)(x₁ - x₂) = (x₁ - x₂)(y₂ - y₁)
AND
Putting these values in (i), we get,
c = ax₁ + by₁
Thus, we now have the values of a, b, and c which means that we have the line in the coordinate plane.

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

string getLine(int x1, int y1, int x2, int y2)
{

    // Calculate numerator and denominator of the slope
    int dy = y2 - y1;
    int dx = x2 - x1;

    // Using point-slope form:
    // y - y1 = (dy / dx) * (x - x1)

    // Cross multiply and rearrange:
    // dx(y - y1) = dy(x - x1)
    // dy*x - dx*y = dy*x1 - dx*y1

    int a = dy;
    int b = -dx;
    int c = dy * x1 - dx * y1;

    string res;

    // Construct the equation
    if (b >= 0)
        res = to_string(a) + "x+" + to_string(b) + "y=" + to_string(c);
    else
        res = to_string(a) + "x" + to_string(b) + "y=" + to_string(c);

    return res;
}

int main()
{
    int x1 = 3, y1 = 2, x2 = 5, y2 = 7;

    cout << getLine(x1, y1, x2, y2);

    return 0;
}

public class Main {

    public static String getLine(int x1, int y1, int x2, int y2) {

        // Calculate numerator and denominator of the slope
        int dy = y2 - y1;
        int dx = x2 - x1;

        // Using point-slope form:
        // y - y1 = (dy / dx) * (x - x1)

        // Cross multiply and rearrange:
        // dx(y - y1) = dy(x - x1)
        // dy*x - dx*y = dy*x1 - dx*y1

        int a = dy;
        int b = -dx;
        int c = dy * x1 - dx * y1;

        String res;

        // Construct the equation
        if (b >= 0)
            res = a + "x+" + b + "y=" + c;
        else
            res = a + "x" + b + "y=" + c;

        return res;
    }

    public static void main(String[] args) {
        int x1 = 3, y1 = 2, x2 = 5, y2 = 7;

        System.out.println(getLine(x1, y1, x2, y2));
    }
}

def getLine(x1, y1, x2, y2):

    # Calculate numerator and denominator of the slope
    dy = y2 - y1
    dx = x2 - x1

    # Using point-slope form:
    # y - y1 = (dy / dx) * (x - x1)

    # Cross multiply and rearrange:
    # dx(y - y1) = dy(x - x1)
    # dy*x - dx*y = dy*x1 - dx*y1

    a = dy
    b = -dx
    c = dy * x1 - dx * y1

    res = ""  # Construct the equation
    if b >= 0:
        res = f"{a}x+{b}y={c}"
    else:
        res = f"{a}x{b}y={c}"

    return res

x1 = 3
y1 = 2
x2 = 5
y2 = 7

print(getLine(x1, y1, x2, y2))

using System;

class Program
{
    static string getLine(int x1, int y1, int x2, int y2)
    {
        // Calculate numerator and denominator of the slope
        int dy = y2 - y1;
        int dx = x2 - x1;

        // Using point-slope form:
        // y - y1 = (dy / dx) * (x - x1)

        // Cross multiply and rearrange:
        // dx(y - y1) = dy(x - x1)
        // dy*x - dx*y = dy*x1 - dx*y1

        int a = dy;
        int b = -dx;
        int c = dy * x1 - dx * y1;

        string res;

        // Construct the equation
        if (b >= 0)
            res = a.ToString() + "x+" + b.ToString() + "y=" + c.ToString();
        else
            res = a.ToString() + "x" + b.ToString() + "y=" + c.ToString();

        return res;
    }

    static void Main()
    {
        int x1 = 3, y1 = 2, x2 = 5, y2 = 7;

        Console.WriteLine(getLine(x1, y1, x2, y2));
    }
}

function getLine(x1, y1, x2, y2) {

    // Calculate numerator and denominator of the slope
    let dy = y2 - y1;
    let dx = x2 - x1;

    // Using point-slope form:
    // y - y1 = (dy / dx) * (x - x1)

    // Cross multiply and rearrange:
    // dx(y - y1) = dy(x - x1)
    // dy*x - dx*y = dy*x1 - dx*y1

    let a = dy;
    let b = -dx;
    let c = dy * x1 - dx * y1;

    let res;  // Construct the equation
    if (b >= 0)
        res = a + "x+" + b + "y=" + c;
    else
        res = a + "x" + b + "y=" + c;

    return res;
}

let x1 = 3, y1 = 2, x2 = 5, y2 = 7;

console.log(getLine(x1, y1, x2, y2));

5x-2y=11