Given six integers, a, b, c, i, j, and k representing the equation of the circle 
and equation of the line 
, the task is to find the length of the intercept cut off from the given line to the circle.
Examples:
Input: a = 0, b = 0, c = -4, i = 2, j = -1, k = 1
Output: 3.89872Input: a = 5, b = 6, c = -16, i = 1, j = 4, k = 3
Output: 6.9282
Approach: The problem can be solved based on the following mathematical fact.
As the line is intercepted by the circle the intercepted segment forms a chord of the circle. Now if a perpendicular is drawn from the centre of the circle to a chord, it divides the chord into two equal parts.
We can easily calculate the radius of the circle and the perpendicular distance of the chord from the centre from the given equations Using them we can get the length of the intercepted line segment.
Follow the steps below to solve the problem:
- Find the center of the circle, say
as 
and 
. - The perpendicular from the center divides the intercept into two equal parts, therefore calculate the length of one of the parts and multiply it by 2 to get the total length of the intercept.
- Calculate the value of radius (r) using the formula:
, where 
and 
- Calculate the value of perpendicular distance ( d ) of center O from the line by using the formula:
- Now from the Pythagoras theorem in triangle OCA:
as 
and 
.
, where 
and 
- After completing the above steps, print the value of twice of AC to get the length of the total intercept.
Below is the implementation of the above approach.
C++
// C++ program for the above approach#include <bits/stdc++.h>using namespace std;// Function to find the// radius of a circledouble radius(int a, int b, int c){ // g and f are the coordinates // of the center int g = a / 2; int f = b / 2; // Case of invalid circle if (g * g + f * f - c < 0) return (-1); // Apply the radius formula return (sqrt(g * g + f * f - c));}// Function to find the perpendicular// distance between circle center and the linedouble centerDistanceFromLine(int a, int b, int i, int j, int k){ // Store the coordinates of center int g = a / 2; int f = b / 2; // Stores the perpendicular distance // between the line and the point double distance = fabs(i * g + j * f + k) / (sqrt(i * i + j * j)); // Invalid Case if (distance < 0) return (-1); // Return the distance return distance;}// Function to find the length of intercept// cut off from a line by a circlevoid interceptLength(int a, int b, int c, int i, int j, int k){ // Calculate the value of radius double rad = radius(a, b, c); // Calculate the perpendicular distance // between line and center double dist = centerDistanceFromLine(a, b, i, j, k); // Invalid Case if (rad < 0 || dist < 0) { cout << "circle not possible"; return; } // If line do not cut circle if (dist > rad) { cout << "Line not cutting circle"; } // Print the intercept length else cout << 2 * sqrt(rad * rad - dist * dist);}// Driver Codeint main(){ // Given Input int a = 0, b = 0, c = -4; int i = 2, j = -1, k = 1; // Function Call interceptLength(a, b, c, i, j, k); return 0;} |
Java
// Java program for the above approachclass GFG{// Function to find the// radius of a circlestatic double radius(int a, int b, int c){ // g and f are the coordinates // of the center int g = a / 2; int f = b / 2; // Case of invalid circle if (g * g + f * f - c < 0) return (-1); // Apply the radius formula return (Math.sqrt(g * g + f * f - c));}// Function to find the perpendicular// distance between circle center and the linestatic double centerDistanceFromLine(int a, int b, int i, int j, int k){ // Store the coordinates of center int g = a / 2; int f = b / 2; // Stores the perpendicular distance // between the line and the point double distance = Math.abs(i * g + j * f + k) / (Math.sqrt(i * i + j * j)); // Invalid Case if (distance < 0) return (-1); // Return the distance return distance;}// Function to find the length of intercept// cut off from a line by a circlestatic void interceptLength(int a, int b, int c, int i, int j, int k){ // Calculate the value of radius double rad = radius(a, b, c); // Calculate the perpendicular distance // between line and center double dist = centerDistanceFromLine( a, b, i, j, k); // Invalid Case if (rad < 0 || dist < 0) { System.out.println("circle not possible"); return; } // If line do not cut circle if (dist > rad) { System.out.println("Line not cutting circle"); } // Print the intercept length else System.out.println(2 * Math.sqrt( rad * rad - dist * dist));}// Driver codepublic static void main(String[] args){ // Given Input int a = 0, b = 0, c = -4; int i = 2, j = -1, k = 1; // Function Call interceptLength(a, b, c, i, j, k);}}// This code is contributed by abhinavjain194 |
Python3
# Python3 program for the above approachimport math# Function to find the# radius of a circledef radius(a, b, c): # g and f are the coordinates # of the center g = a / 2 f = b / 2 # Case of invalid circle if (g * g + f * f - c < 0): return(-1) # Apply the radius formula return(math.sqrt(g * g + f * f - c))# Function to find the perpendicular# distance between circle center and the linedef centerDistanceFromLine(a, b, i, j, k): # Store the coordinates of center g = a / 2 f = b / 2 # Stores the perpendicular distance # between the line and the point distance = (abs(i * g + j * f + k) / (math.sqrt(i * i + j * j))) # Invalid Case if (distance < 0): return (-1) # Return the distance return distance# Function to find the length of intercept# cut off from a line by a circledef interceptLength(a, b, c, i, j, k): # Calculate the value of radius rad = radius(a, b, c) # Calculate the perpendicular distance # between line and center dist = centerDistanceFromLine( a, b, i, j, k) # Invalid Case if (rad < 0 or dist < 0): print("circle not possible") return # If line do not cut circle if (dist > rad): print("Line not cutting circle") # Print the intercept length else: print(2 * math.sqrt( rad * rad - dist * dist))# Driver Codeif __name__ == "__main__": # Given Input a = 0 b = 0 c = -4 i = 2 j = -1 k = 1 # Function Call interceptLength(a, b, c, i, j, k)# This code is contributed by ukasp |
C#
// C# program for the above approachusing System;class GFG{ // Function to find the// radius of a circlestatic double radius(int a, int b, int c){ // g and f are the coordinates // of the center int g = a / 2; int f = b / 2; // Case of invalid circle if (g * g + f * f - c < 0) return (-1); // Apply the radius formula return(Math.Sqrt(g * g + f * f - c));} // Function to find the perpendicular// distance between circle center and the linestatic double centerDistanceFromLine(int a, int b, int i, int j, int k){ // Store the coordinates of center int g = a / 2; int f = b / 2; // Stores the perpendicular distance // between the line and the point double distance = Math.Abs(i * g + j * f + k) / (Math.Sqrt(i * i + j * j)); // Invalid Case if (distance < 0) return (-1); // Return the distance return distance;} // Function to find the length of intercept// cut off from a line by a circlestatic void interceptLength(int a, int b, int c, int i, int j, int k){ // Calculate the value of radius double rad = radius(a, b, c); // Calculate the perpendicular distance // between line and center double dist = centerDistanceFromLine( a, b, i, j, k); // Invalid Case if (rad < 0 || dist < 0) { Console.WriteLine("circle not possible"); return; } // If line do not cut circle if (dist > rad) { Console.WriteLine("Line not cutting circle"); } // Print the intercept length else Console.WriteLine(2 * Math.Sqrt( rad * rad - dist * dist));}// Driver codepublic static void Main(String []args){ // Given Input int a = 0, b = 0, c = -4; int i = 2, j = -1, k = 1; // Function Call interceptLength(a, b, c, i, j, k);}}// This code is contributed by sanjoy_62 |
Javascript
<script> // JavaScript program for the above approach // Function to find the // radius of a circle function radius(a, b, c) { // g and f are the coordinates // of the center let g = a / 2; let f = b / 2; // Case of invalid circle if (g * g + f * f - c < 0) return (-1); // Apply the radius formula return (Math.sqrt(g * g + f * f - c)); } // Function to find the perpendicular // distance between circle center and the line function centerDistanceFromLine(a, b, i, j, k) { // Store the coordinates of center let g = a / 2; let f = b / 2; // Stores the perpendicular distance // between the line and the point let distance = Math.abs(i * g + j * f + k) / (Math.sqrt(i * i + j * j)); // Invalid Case if (distance < 0) return (-1); // Return the distance return distance; } // Function to find the length of intercept // cut off from a line by a circle function interceptLength(a, b, c, i, j, k) { // Calculate the value of radius let rad = radius(a, b, c); // Calculate the perpendicular distance // between line and center let dist = centerDistanceFromLine( a, b, i, j, k); // Invalid Case if (rad < 0 || dist < 0) { document.write("circle not possible"); return; } // If line do not cut circle if (dist > rad) { document.write("Line not cutting circle"); } // Print the intercept length else document.write(2 * Math.sqrt( rad * rad - dist * dist)); } // Driver code // Given Input let a = 0, b = 0, c = -4; let i = 2, j = -1, k = 1; // Function Call interceptLength(a, b, c, i, j, k); // This code is contributed by Hritik </script> |
Output
3.89872
Time Complexity: O(logN), for using in-built sqrt() function.
Auxiliary Space: O(1)
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