# Equation of a straight line with perpendicular distance D from origin and an angle A between the perpendicular from origin and x-axis

Given two integers **D** and **A** representing the perpendicular distance from the origin to a straight line and the angle made by the perpendicular with the positive **x**-axis respectively, the task is to find the equation of the straight line.

**Examples:**

Input:D = 10, A = 30 degreesOutput:0.87x +0.50y = 10

Input:D = 12, A = 45 degreesOutput:0.71x +0.71y = 12

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

- Let the perpendicular distance be
**(p)**and the angle between the perpendicular and the positive x-axis be**(Î±) degrees**. - Consider a point
**P**with coordinates**(x, y)**on the required line. - Draw a perpendicular from
**P**to meet the x-axis at**L**. - From
**L**, draw a perpendicular on**OQ**at**M**. - Now, draw a perpendicular from
**P**to meet**ML**at**N**.

Now consider right triangle

OLM

â€” (1)Now consider right triangle

PNL

â€” (2)Now

Using equations (1) and (2)

which is the equation of the required line

Below is the implementation of the above approach :

## C++

`// C++ program for the approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `// Function to find equation of a line whose` `// distance from origin and angle made by the` `// perpendicular from origin with x-axis is given` `void` `findLine(` `int` `distance, ` `float` `degree)` `{` ` ` `// Convert angle from degree to radian` ` ` `float` `x = degree * 3.14159 / 180;` ` ` `// Handle the special case` ` ` `if` `(degree > 90) {` ` ` `cout << ` `"Not Possible"` `;` ` ` `return` `;` ` ` `}` ` ` `// Calculate the sin and cos of angle` ` ` `float` `result_1 = ` `sin` `(x);` ` ` `float` `result_2 = ` `cos` `(x);` ` ` `// Print the equation of the line` ` ` `cout << fixed << setprecision(2)` ` ` `<< result_2 << ` `"x +"` ` ` `<< result_1 << ` `"y = "` `<< distance;` `}` `// Driver Code` `int` `main()` `{` ` ` `// Given Input` ` ` `int` `D = 10;` ` ` `float` `A = 30;` ` ` `// Function Call` ` ` `findLine(D, A);` ` ` `return` `0;` `}` |

## Java

`// Java program for the approach` `class` `GFG{` `// Function to find equation of a line whose` `// distance from origin and angle made by the` `// perpendicular from origin with x-axis is given` `static` `void` `findLine(` `int` `distance, ` `float` `degree)` `{` ` ` `// Convert angle from degree to radian` ` ` `float` `x = (` `float` `) (degree * ` `3.14159` `/ ` `180` `);` ` ` `// Handle the special case` ` ` `if` `(degree > ` `90` `) {` ` ` `System.out.print(` `"Not Possible"` `);` ` ` `return` `;` ` ` `}` ` ` `// Calculate the sin and cos of angle` ` ` `float` `result_1 = (` `float` `) Math.sin(x);` ` ` `float` `result_2 = (` `float` `) Math.cos(x);` ` ` `// Print the equation of the line` ` ` `System.out.print(String.format(` `"%.2f"` `,result_2)+ ` `"x +"` ` ` `+ String.format(` `"%.2f"` `,result_1)+ ` `"y = "` `+ distance);` `}` `// Driver Code` `public` `static` `void` `main(String[] args)` `{` ` ` `// Given Input` ` ` `int` `D = ` `10` `;` ` ` `float` `A = ` `30` `;` ` ` `// Function Call` ` ` `findLine(D, A);` `}` `}` `// This code is contributed by shikhasingrajput` |

## Python3

`# Python3 program for the approach` `import` `math` `# Function to find equation of a line whose` `# distance from origin and angle made by the` `# perpendicular from origin with x-axis is given` `def` `findLine(distance, degree):` ` ` `# Convert angle from degree to radian` ` ` `x ` `=` `degree ` `*` `3.14159` `/` `180` ` ` ` ` `# Handle the special case` ` ` `if` `(degree > ` `90` `):` ` ` `print` `(` `"Not Possible"` `)` ` ` `return` ` ` ` ` `# Calculate the sin and cos of angle` ` ` `result_1 ` `=` `math.sin(x)` ` ` `result_2 ` `=` `math.cos(x)` ` ` ` ` `# Print the equation of the line` ` ` `print` `(` `'%.2f'` `%` `result_2,` ` ` `"x +"` `, ` `'%.2f'` `%` `result_1,` ` ` `"y = "` `, distance, sep ` `=` `"")` `# Driver code` `# Given Input` `D ` `=` `10` `A ` `=` `30` `# Function Call` `findLine(D, A)` `# This code is contributed by mukesh07` |

## C#

`// C# program for the approach` `using` `System;` `class` `GFG` `{` ` ` ` ` `// Function to find equation of a line whose` ` ` `// distance from origin and angle made by the` ` ` `// perpendicular from origin with x-axis is given` ` ` `static` `void` `findLine(` `int` `distance, ` `float` `degree)` ` ` `{` ` ` `// Convert angle from degree to radian` ` ` `float` `x = (` `float` `)(degree * 3.14159 / 180);` ` ` ` ` `// Handle the special case` ` ` `if` `(degree > 90) {` ` ` `Console.WriteLine(` `"Not Possible"` `);` ` ` `return` `;` ` ` `}` ` ` ` ` `// Calculate the sin and cos of angle` ` ` `float` `result_1 = (` `float` `)(Math.Sin(x));` ` ` `float` `result_2 = (` `float` `)(Math.Cos(x));` ` ` ` ` `// Print the equation of the line` ` ` `Console.WriteLine(result_2.ToString(` `"0.00"` `) + ` `"x +"` ` ` `+ result_1.ToString(` `"0.00"` `) + ` `"y = "` `+ distance);` ` ` `}` ` ` `static` `void` `Main ()` ` ` `{` ` ` `// Given Input` ` ` `int` `D = 10;` ` ` `float` `A = 30;` ` ` ` ` `// Function Call` ` ` `findLine(D, A);` ` ` `}` `}` `// This code is contributed by suresh07.` |

## Javascript

`<script>` ` ` ` ` `// JavaScript program for the above approach` ` ` `// Function to find equation of a line whose` ` ` `// distance from origin and angle made by the` ` ` `// perpendicular from origin with x-axis is given` ` ` `function` `findLine(distance, degree) {` ` ` `// Convert angle from degree to radian` ` ` `let x = degree * 3.14159 / 180;` ` ` `// Handle the special case` ` ` `if` `(degree > 90) {` ` ` `document.write(` `"Not Possible"` `);` ` ` `return` `;` ` ` `}` ` ` `// Calculate the sin and cos of angle` ` ` `let result_1 = Math.sin(x);` ` ` `let result_2 = Math.cos(x);` ` ` `// Print the equation of the line` ` ` `document.write(result_2.toPrecision(2) + ` `"x + "` ` ` `+ result_1.toPrecision(2) + ` `"y = "` `+ distance);` ` ` `}` ` ` `// Driver Code` ` ` `// Given Input` ` ` `let D = 10;` ` ` `let A = 30;` ` ` `// Function Call` ` ` `findLine(D, A);` ` ` `// This code is contributed by Hritik` ` ` `</script>` |

**Output:**

0.87x +0.50y = 10

**Time Complexity:** O(1)**Auxiliary Space:** O(1)