Equation of a normal to a Circle from a given point

Given three integers a, b, c representing coefficients of the equation x² + y² + ax + by + c = 0 of a circle, the task is to find the equation of the normal to the circle from a given point (x₁, y₁).
Note: Normal is a line perpendicular to the tangent at the point of contact between the tangent and the curve.

Examples:

Input: a = 4, b = 6, c = 5, x₁ = 12, y₁ = 14 
Output: y = 1.1x + 0.8

Input: a = 6, b = 12, c = 5, x₁ = 9, y₁ = 3 
Output: y = -0.5x + 7.5

Approach: Follow the steps below to solve the problem:

• The normal to a circle passes through the center of the circle.
• Therefore, find the coordinates of the center of the circle (g, f), where g = a/2 and f = b/2.
• Since the center of the circle and the point where the normal is drawn lie on the normal, calculate the slope of the normal (m) as m = (y₁ – f) / (x₁ – g).
• Hence, the equation of the normal is y – y₁ = m * (x – x₁).

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 the slope
double normal_slope(double a, double b,
                    double x1, double y1)
{
    // Store the coordinates of
    // the center of the circle
    double g = a / 2;
    double f = b / 2;

    // If slope becomes infinity
    if (g - x1 == 0)
        return (-1);

    // Stores the slope
    double slope = (f - y1) / (g - x1);

    // If slope is zero
    if (slope == 0)
        return (-2);

    // Return the result
    return slope;
}

// Function to find the equation of the
// normal to a circle from a given point
void normal_equation(double a, double b,
                     double x1, double y1)
{
    // Stores the slope of the normal
    double slope = normal_slope(a, b, x1, y1);

    // If slope becomes infinity
    if (slope == -1) {
        cout << "x = " << x1;
    }

    // If slope is zero
    if (slope == -2) {
        cout << "y = " << y1;
    }

    // Otherwise, print the
    // equation of the normal
    if (slope != -1 && slope != -2) {

        x1 *= -slope;
        x1 += y1;

        if (x1 > 0)
            cout << "y = " << slope
                 << "x + " << x1;
        else
            cout << "y = " << slope
                 << "x " << x1;
    }
}

// Driver Code
int main()
{
    // Given Input
    int a = 4, b = 6, c = 5;
    int x1 = 12, y1 = 14;

    // Function Call
    normal_equation(a, b, x1, y1);

    return 0;
}

Java

// Java program for the above approach
import java.util.*;

class GFG{

// Function to calculate the slope
static double normal_slope(double a, double b,
                           double x1, double y1)
{

    // Store the coordinates of
    // the center of the circle
    double g = a / 2;
    double f = b / 2;

    // If slope becomes infinity
    if (g - x1 == 0)
        return (-1);

    // Stores the slope
    double slope = (f - y1) / (g - x1);

    // If slope is zero
    if (slope == 0)
        return (-2);

    // Return the result
    return slope;
}

// Function to find the equation of the
// normal to a circle from a given point
static void normal_equation(double a, double b,
                            double x1, double y1)
{

    // Stores the slope of the normal
    double slope = normal_slope(a, b, x1, y1);

    // If slope becomes infinity
    if (slope == -1)
    {
        System.out.print("x = " +  x1);
    }

    // If slope is zero
    if (slope == -2)
    {
        System.out.print("y = " +  y1);
    }

    // Otherwise, print the
    // equation of the normal
    if (slope != -1 && slope != -2)
    {
        x1 *= -slope;
        x1 += y1;

        if (x1 > 0)
            System.out.print("y = " +  slope +
                             "x + " +  x1);
        else
            System.out.print("y = " +  slope +
                             "x " +  x1);
    }
}

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

    // Given Input
    int a = 4, b = 6;
    int x1 = 12, y1 = 14;

    // Function Call
    normal_equation(a, b, x1, y1);
}
}

// This code is contributed by 29AjayKumar

Python3

# Python3 program for the above approach

# Function to calculate the slope
def normal_slope(a, b, x1, y1):

    # Store the coordinates
    # the center of the circle
    g = a / 2
    f = b / 2

    # If slope becomes infinity
    if (g - x1 == 0):
        return (-1)

    # Stores the slope
    slope = (f - y1) / (g - x1)

    # If slope is zero
    if (slope == 0):
        return (-2)

    # Return the result
    return slope

# Function to find the equation of the
# normal to a circle from a given point
def normal_equation(a, b, x1, y1):

    # Stores the slope of the normal
    slope = normal_slope(a, b, x1, y1)

    # If slope becomes infinity
    if (slope == -1) :
        print("x = ", x1)

    # If slope is zero
    if (slope == -2) :
        print("y = ", y1)

    # Otherwise, print the
    # equation of the normal
    if (slope != -1 and slope != -2):

        x1 *= -slope
        x1 += y1

        if (x1 > 0) :
            print("y = ", slope, "x + ", x1)
        else :
            print("y = ", slope, "x ", x1)

# Driver Code

# Given Input
a = 4
b = 6
c = 5
x1 = 12
y1 = 14

# Function Call
normal_equation(a, b, x1, y1)

# This code is contributed by Dharanendra L V.

C#

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

class GFG{

// Function to calculate the slope
static double normal_slope(double a, double b,
                           double x1, double y1)
{

    // Store the coordinates of
    // the center of the circle
    double g = a / 2;
    double f = b / 2;

    // If slope becomes infinity
    if (g - x1 == 0)
        return (-1);

    // Stores the slope
    double slope = (f - y1) / (g - x1);

    // If slope is zero
    if (slope == 0)
        return (-2);

    // Return the result
    return slope;
}

// Function to find the equation of the
// normal to a circle from a given point
static void normal_equation(double a, double b,
                            double x1, double y1)
{

    // Stores the slope of the normal
    double slope = normal_slope(a, b, x1, y1);

    // If slope becomes infinity
    if (slope == -1)
    {
         Console.WriteLine( "x = " + x1);
    }

    // If slope is zero
    if (slope == -2)
    {
         Console.WriteLine("y = " + y1);
    }

    // Otherwise, print the
    // equation of the normal
    if (slope != -1 && slope != -2)
    {
        x1 *= -slope;
        x1 += y1;

        if (x1 > 0)
            Console.WriteLine("y = " + slope +
                              "x +" + Math.Round(x1, 2));
        else
            Console.WriteLine("y = " + slope +
                              "x " + Math.Round(x1, 2));
    }
}

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

    // Given Input
    int a = 4, b = 6;
    //int c = 5;

    int x1 = 12, y1 = 14;

    // Function Call
    normal_equation(a, b, x1, y1);
}
}

// This code is contributed by sanjoy_62

Javascript

<script>

// JavaScript program for the above approach


// Function to calculate the slope
function normal_slope( a,  b, x1,  y1)
{
    // Store the coordinates of
    // the center of the circle
    var g = a / 2;
    var f = b / 2;

    // If slope becomes infinity
    if (g - x1 == 0)
        return (-1);

    // Stores the slope
    var slope = (f - y1) / (g - x1);

    // If slope is zero
    if (slope == 0)
        return (-2);

    // Return the result
    return slope;
}

// Function to find the equation of the
// normal to a circle from a given point

function normal_equation( a,  b, x1,  y1)
{
    // Stores the slope of the normal
    var slope = normal_slope(a, b, x1, y1);

    // If slope becomes infinity
    if (slope == -1) {
        document.write("x = " + x1);
    }

    // If slope is zero
    if (slope == -2) {
        document.write("y = " + y1);
    }

    // Otherwise, print the
    // equation of the normal
    if (slope != -1 && slope != -2) {

        x1 *= -slope;
        x1 += y1;

        if (x1 > 0)
            document.write(
            "y = " + slope + "x + " + x1.toFixed(1)
            );
        else
             document.write(
             "y = "  + slope+ "x " + x1.toFixed(1)
             );
    }
}

// Driver Code

    // Given Input
    var a = 4, b = 6, c = 5;
    var x1 = 12, y1 = 14;

    // Function Call
    normal_equation(a, b, x1, y1);


</script>

Output:

y = 1.1x + 0.8

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