Check if a line touches or intersects a circle

Given coordinate of the center and radius > 1 of a circle and the equation of a line. The task is to check if the given line collides with the circle or not. There are three possibilities : 

1. Line intersects the circle.
2. Line touches the circle.
3. Line is outside the circle
    

Note: General equation of a line is a*x + b*y + c = 0, so only constant a, b, c are given in the input.

Examples : 

Input : radius = 5, center = (0, 0), 
        a = 1, b = -1, c = 0.
Output : Intersect
Input :  radius = 5, center = (0, 0), 
         a = 5, b = 0, c = 0.
Output : Intersect
Input : radius = 5, center = (0, 0),
         a = 1, b = 1, c = -16.
Output : Outside

The idea is to compare the perpendicular distance between center of circle and line with the radius of the circle.
Algorithm: 
1. Find the perpendicular (say p) between center of circle and given line. 
2. Compare this distance p with radius r. 
......a) If p > r, then line lie outside the circle. 
......b) If p = r, then line touches the circle. 
......c) If p < r, then line intersect the circle.
How to find the perpendicular distance? 
Distance of a line from a point can be computed using below formula:
\frac{ax_{0}+by_{0}+c}{\sqrt{a^{2}+b^{2}}}      

// CPP program to check if a line touches or 
// intersects or outside a circle.
#include <bits/stdc++.h>
using namespace std;

void checkCollision(int a, int b, int c, 
                  int x, int y, int radius)
{
    // Finding the distance of line from center.
    int dist = (abs(a * x + b * y + c)) / 
                     sqrt(a * a + b * b);

    // Checking if the distance is less than, 
    // greater than or equal to radius.
    if (radius == dist)
        cout << "Touch" << endl;
    else if (radius > dist)
        cout << "Intersect" << endl;
    else
        cout << "Outside" << endl;
}

// Driven Program
int main()
{
    int radius = 5;
    int x = 0, y = 0;
    int a = 3, b = 4, c = 25;
    checkCollision(a, b, c, x, y, radius);
    return 0;
}

// Java program to check if a line touches or 
// intersects or outside a circle.

import java.io.*;

class GFG {
    
    static void checkCollision(int a, int b, int c, 
                               int x, int y, int radius)
    {
        // Finding the distance of line from center.
        double dist = (Math.abs(a * x + b * y + c)) / 
                        Math.sqrt(a * a + b * b);
    
        // Checking if the distance is less than, 
        // greater than or equal to radius.
        if (radius == dist)
            System.out.println ( "Touch" );
        else if (radius > dist)
            System.out.println( "Intersect") ;
        else
            System.out.println( "Outside") ;
    }
    
    // Driven Program
    public static void main (String[] args) 
    {
        int radius = 5;
        int x = 0, y = 0;
        int a = 3, b = 4, c = 25;
        checkCollision(a, b, c, x, y, radius);
    
    }
}

// 

# python program to check if a line
# touches or  intersects or outside
# a circle.

import math

def checkCollision(a, b, c, x, y, radius):
    
    # Finding the distance of line 
    # from center.
    dist = ((abs(a * x + b * y + c)) /
            math.sqrt(a * a + b * b))

    # Checking if the distance is less 
    # than, greater than or equal to radius.
    if (radius == dist):
        print("Touch")
    elif (radius > dist):
        print("Intersect")
    else:
        print("Outside")

# Driven Program
radius = 5
x = 0
y = 0
a = 3
b = 4
c = 25
checkCollision(a, b, c, x, y, radius)

# This code is contributed by Sam007

// C# program to check if a line touches or 
// intersects or outside a circle.
using System;

class GFG {
    
    static void checkCollision(int a, int b, int c, 
                            int x, int y, int radius)
    {
        // Finding the distance of line from center.
        double dist = (Math.Abs(a * x + b * y + c)) / 
                        Math.Sqrt(a * a + b * b);
    
        // Checking if the distance is less than, 
        // greater than or equal to radius.
        if (radius == dist)
            Console.WriteLine ("Touch");
        else if (radius > dist)
            Console.WriteLine("Intersect");
        else
            Console.WriteLine("Outside");
    }
    
    // Driven Program
    public static void Main () 
    {
        int radius = 5;
        int x = 0, y = 0;
        int a = 3, b = 4, c = 25;
        
        checkCollision(a, b, c, x, y, radius);
    
    }
}

// This article is contributed by vt_m.

<script>

// JavaScript program to check if a line touches or 
// intersects or outside a circle.

function checkCollision(a, b, c, x, y, radius)
    {
    
        // Finding the distance of line from center.
        let dist = (Math.abs(a * x + b * y + c)) / 
                        Math.sqrt(a * a + b * b);
      
        // Checking if the distance is less than, 
        // greater than or equal to radius.
        if (radius == dist)
            document.write ( "Touch" );
        else if (radius > dist)
            document.write( "Intersect") ;
        else
            document.write( "Outside") ;
    }

// Driver Code
        let radius = 5;
        let x = 0, y = 0;
        let a = 3, b = 4, c = 25;
        checkCollision(a, b, c, x, y, radius);

// This code is contributed by susmitakundugoaldanga.
</script>

<?php
// PHP program to check if a line  
// touches or intersects or outside 
// a circle.

function checkCollision($a, $b, $c, 
                        $x, $y, $radius)
{
    // Finding the distance 
    // of line from center.
    $dist = (abs($a * $x + $b * $y + $c)) / 
                   sqrt($a * $a + $b * $b);

    // Checking if the distance is less than, 
    // greater than or equal to radius.
    if ($radius == $dist)
        echo "Touch";
    else if ($radius > $dist)
        echo "Intersect";
    else
        echo "Outside" ;
}

// Driver Code
$radius = 5;
$x = 0;
$y = 0;
$a = 3;
$b = 4;
$c = 25;
checkCollision($a, $b, $c, $x, $y, $radius);

// This code is contributed by Sam007
?>

Touch

Time Complexity : O(log(a*a + b*b)) as it is using inbuilt sqrt function
Auxiliary Space : O(1)

This article is contributed by Anuj Chauhan.