Dada una ecuación de tres líneas,
a 1 x + b 1 y + c 1 = 0
a 2 x + b 2 y + c 2 = 0
a 3 x + b 3 y + c 3 = 0
La tarea es verificar si los tres dados las líneas son concurrentes o no. Se dice que tres rectas son concurrentes si pasan por un punto, es decir, se encuentran en un punto.
Ejemplos:
Input : a1 = 2, b1 = -3, c1 = 5
a2 = 3, b2 = 4, c2 = -7
a3 = 9, b3 = -5, c3 = 8
Output : Yes
Input : a1 = 2, b1 = -3, c1 = 5
a2 = 3, b2 = 4, c2 = -7
a3 = 9, b3 = -5, c3 = 4
Output : No
Sea
a 1 x + b 1 y + c 1 = 0 ………. (1)
a 2 x + b 2 y + c 2 = 0 ………. (2)
a 3 x + b 3 y + c 3 = 0 ………. (3)
Suponga que la ecuación (i) y (ii) se cruzan en (x 1 , y 1 ). Entonces (x 1 , y 1 ) satisfará ambas ecuaciones.
Por lo tanto, resolviendo (i) y (ii) usando el método de multiplicación cruzada , obtenemos,
(x 1 /b 1 c 2 – b 2 c 1) = (y 1 /c 1 a 2 – c 2 a 1 ) = (1/a 1 b 2 – a 2 b 1 )
Por lo tanto,
x 1 = (b 1 c 2 – b 2 c 1 /a 1 b 2 – a 2 b 1 ) y
y 1 = (c 1 a 2 – c 2 a 1 /a 1 b 2 – a 2 b 1), a 1 b 2 – a 2 b 1 != 0
Por lo tanto, las coordenadas requeridas del punto de intersección de las rectas (i) y (ii) son
(b 1 c 2 – b 2 c 1 /a 1 b 2 – a 2 b 1 , c 1 a 2 – c 2 a 1 /a 1 b 2 – a 2 b 1 )
Para que tres de la línea sean concurrentes, (x 1 , y 1) debe satisfacer la ecuación (iii) también.
Entonces,
a 3 x + b 3 y + c 3 = 0
=> a 3 (b 1 c 2 – b 2 c 1 /a 1 b 2 – a 2 b 1 ) + b 3 (c 1 a 2 – c 2 a 1 /a 1 b 2 – a 2 b 1 ) + c 3 = 0
=> a 3 (b 1 c 2– b 2 c 1 ) + b 3 (c 1 a 2 – c 2 a 1 ) + c 3 (a 1 b 2 – a 2 b 1 ) = 0
Entonces, solo necesitamos verificar si la condición anterior se cumple o no.
A continuación se muestra la implementación de este enfoque:
C++
// CPP Program to check if three straight
// line are concurrent or not
#include <bits/stdc++.h>
using namespace std;
// Return true if three line are concurrent,
// else false.
bool checkConcurrent(int a1, int b1, int c1,
int a2, int b2, int c2,
int a3, int b3, int c3)
{
return (a3 * (b1 * c2 - b2 * c1) +
b3 * (c1 * a2 - c2 * a1) +
c3 * (a1 * b2 - a2 * b1) == 0);
}
// Driven Program
int main()
{
int a1 = 2, b1 = -3, c1 = 5;
int a2 = 3, b2 = 4, c2 = -7;
int a3 = 9, b3 = -5, c3 = 8;
(checkConcurrent(a1, b1, c1, a2, b2, c2,
a3, b3, c3) ? (cout << "Yes") : (cout << "No"));
return 0;
}
Java
// Java Program to check if three straight
// line are concurrent or no
import java.io.*;
class GFG {
// Return true if three line are concurrent,
// else false.
static boolean checkConcurrent(int a1, int b1,
int c1, int a2, int b2, int c2,
int a3, int b3, int c3)
{
return (a3 * (b1 * c2 - b2 * c1) +
b3 * (c1 * a2 - c2 * a1) +
c3 * (a1 * b2 - a2 * b1) == 0);
}
// Driven Program
public static void main (String[] args)
{
int a1 = 2, b1 = -3, c1 = 5;
int a2 = 3, b2 = 4, c2 = -7;
int a3 = 9, b3 = -5, c3 = 8;
if(checkConcurrent(a1, b1, c1, a2, b2,
c2, a3, b3, c3))
System.out.println( "Yes");
else
System.out.println( "No");
}
}
// This code is contributed by anuj_67.
Python 3
# Python3 Program to check if three straight
# line are concurrent or not
# Return true if three line are concurrent,
# else false.
def checkConcurrent(a1, b1, c1, a2, b2, c2,
a3, b3, c3):
return (a3 * (b1 * c2 - b2 * c1) +
b3 * (c1 * a2 - c2 * a1) +
c3 * (a1 * b2 - a2 * b1) == 0)
# Driven Program
a1 = 2
b1 = -3
c1 = 5
a2 = 3
b2 = 4
c2 = -7
a3 = 9
b3 = -5
c3 = 8
if(checkConcurrent(a1, b1, c1, a2, b2, c2,
a3, b3, c3)):
print("Yes")
else:
print("No")
# This code is contributed by Smitha
C#
// C# Program to check if three straight
// line are concurrent or no
using System;
class GFG {
// Return true if three line are concurrent,
// else false.
static bool checkConcurrent(int a1, int b1,
int c1, int a2, int b2, int c2,
int a3, int b3, int c3)
{
return (a3 * (b1 * c2 - b2 * c1) +
b3 * (c1 * a2 - c2 * a1) +
c3 * (a1 * b2 - a2 * b1) == 0);
}
// Driven Program
public static void Main ()
{
int a1 = 2, b1 = -3, c1 = 5;
int a2 = 3, b2 = 4, c2 = -7;
int a3 = 9, b3 = -5, c3 = 8;
if(checkConcurrent(a1, b1, c1, a2, b2,
c2, a3, b3, c3))
Console.WriteLine( "Yes");
else
Console.WriteLine( "No");
}
}
// This code is contributed by anuj_67.
PHP
<?php
// PHP Program to check if three straight
// line are concurrent or not
// Return true if three line are
// concurrent, else false.
function checkConcurrent($a1, $b1, $c1,
$a2, $b2, $c2,
$a3, $b3, $c3)
{
return ($a3 * ($b1 * $c2 - $b2 * $c1) +
$b3 * ($c1 * $a2 - $c2 * $a1) +
$c3 * ($a1 * $b2 - $a2 * $b1) == 0);
}
// Driver Code
$a1 = 2; $b1 = -3; $c1 = 5;
$a2 = 3; $b2 = 4; $c2 = -7;
$a3 = 9; $b3 = -5; $c3 = 8;
if(checkConcurrent($a1, $b1, $c1, $a2, $b2,
$c2, $a3, $b3, $c3))
echo "Yes";
else
echo "No";
// This code is contributed by anuj_67.
?>
Javascript
<script>
// Javascript Program to check if three straight
// line are concurrent or not
// Return true if three line are concurrent,
// else false.
function checkConcurrent( a1, b1, c1,
a2, b2, c2,
a3, b3, c3)
{
return (a3 * (b1 * c2 - b2 * c1) +
b3 * (c1 * a2 - c2 * a1) +
c3 * (a1 * b2 - a2 * b1) == 0);
}
// Driven Program
a1 = 2, b1 = -3, c1 = 5;
a2 = 3, b2 = 4, c2 = -7;
a3 = 9, b3 = -5, c3 = 8;
(checkConcurrent(a1, b1, c1, a2, b2, c2,
a3, b3, c3) ? (document.write( "Yes")) : (document.write( "No")));
</script>
Yes