我们给定了抛物线的焦点（x，y）和准线（ax + by + c），我们必须使用它的焦点和准线来找到抛物线的方程。
例子 ：
Input: x1 = 0, y1 = 0, a = 2, b = 1, c = 2
Output: equation of parabola is 16.0 x^2 + 9.0 y^2 + -12.0 x + 16.0 y + 24.0 xy + -4.0 = 0.
Input: x1 = -1, y1 = -2, a = 1, b = -2, c = 3
Output:equation of parabola is 4.0 x^2 + 1.0 y^2 + 4.0 x + 32.0 y + 4.0 xy + 16.0 = 0.
设 P(x, y) 是抛物线上的任何点，其焦点 S(x1, y1) 和准线是直线 ax + by + c =0。
在准线上从 P 垂直绘制 PM。然后根据定义 pf 抛物线距离 SP = PM
SP^2 = PM^2
(x - x1)^2 + (y - y1)^2 = ( ( a*x + b*y + c ) / (sqrt( a*a + b*b )) ) ^ 2// 让 ( a*a + b*b ) = t
x^2 + x1^2 - 2*x1*x + y^2 + y1^2 - 2*y1*y = ( ( a*x + b*y + c ) ^ 2 )/ t在上面的交叉乘法中我们得到
t*x^2 + t*x1^2 - 2*t*x1*x + t*y^2 + t*y1^2 - 2*t*y1*y = ( ( a*x + b*y + c ) ^ 2 )
t*x^2 + t*x1^2 - 2*t*x1*x + t*y^2 + t*y1^2 - 2*t*y1*y = a^2*x^2 + b^2*y^2 + 2*a*x*b*y + c^2 + 2*c*(a*x + b*y)
t*x^2 + t*x1^2 - 2*t*x1*x + t*y^2 + t*y1^2 - 2*t*y1*y = a^2*x^2 + b^2*y^2 + 2*a*x*b*y + c^2 + 2*c*a*x + 2*c*b*y
t*x^2 - a^2*x^2 + t*y^2 - b^2*y^2 - 2*t*x1*x - 2*c*a*x - 2*t*y1*y - 2*c*b*y - 2*a*x*b*y - c^2 + t*x1^2 + t*y1^2 =0.这可以与一般形式进行比较，即
a*x^2 + 2*h*x*y + b*y^2 + 2*g*x + 2*f*y + c = 0.下面是上面的实现：
C++
// C++ program to find equation of a parbola
// using focus and directrix.
#include
#include
#include
#include
using namespace std;
// Function to find equation of parabola.
void equation_parabola(float x1, float y1,
float a, float b, float c)
{
float t = a * a + b * b;
float a1 = t - (a * a);
float b1 = t - (b * b);
float c1 = (-2 * t * x1) - (2 * c * a);
float d1 = (-2 * t * y1) - (2 * c * b);
float e1 = -2 * a * b;
float f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1);
std::cout << std::fixed;
std::cout << std::setprecision(1);
cout << "equation of parabola is " << a1
<< " x^2 + " << b1 << " y^2 + "
<< c1 << " x + " << d1 << " y + "
<< e1 << " xy + " << f1 << " = 0.";
}
// Driver Code
int main()
{
float x1 = 0;
float y1 = 0;
float a = 3;
float b = -4;
float c = 2;
equation_parabola(x1, y1, a, b, c);
return 0;
}
// This code is contributed by Amber_Saxena. Java
// Java program to find equation of a parbola
// using focus and directrix.
import java.util.*;
class solution
{
//Function to find equation of parabola.
static void equation_parabola(float x1, float y1,
float a, float b, float c)
{
float t = a * a + b * b;
float a1 = t - (a * a);
float b1 = t - (b * b);
float c1 = (-2 * t * x1) - (2 * c * a);
float d1 = (-2 * t * y1) - (2 * c * b);
float e1 = -2 * a * b;
float f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1);
System.out.println( "equation of parabola is "+ a1+
" x^2 + " +b1 +" y^2 + "+
c1 + " x + " +d1 + " y + "
+ e1+" xy + " + f1 +" = 0.");
}
// Driver Code
public static void main(String arr[])
{
float x1 = 0;
float y1 = 0;
float a = 3;
float b = -4;
float c = 2;
equation_parabola(x1, y1, a, b, c);
}
}Python3
# Python3 program to find equation of a parbola
# using focus and directrix.
# Function to find equation of parabola.
def equation_parabola(x1, y1, a, b, c) :
t = a * a + b * b
a1 = t - (a * a)
b1 = t - (b * b);
c1 = (-2 * t * x1) - (2 * c * a)
d1 = (-2 * t * y1) - (2 * c * b)
e1 = -2 * a * b
f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1)
print("equation of parabola is", a1 ,"x^2 +" ,b1,
"y^2 +",c1,"x +", d1,"y + ",e1 ,"xy +",f1,"= 0.")
# Driver Code
if __name__ == "__main__" :
x1, y1, a, b, c = 0,0,3,-4,2
equation_parabola(x1, y1, a, b, c)
# This code is contributed by RyugaC#
// C# program to find equation of a parbola
// using focus and directrix.
using System;
class solution
{
//Function to find equation of parabola.
static void equation_parabola(float x1, float y1,
float a, float b, float c)
{
float t = a * a + b * b;
float a1 = t - (a * a);
float b1 = t - (b * b);
float c1 = (-2 * t * x1) - (2 * c * a);
float d1 = (-2 * t * y1) - (2 * c * b);
float e1 = -2 * a * b;
float f1 = (-c * c) + (t * x1 * x1) + (t * y1 * y1);
Console.WriteLine( "equation of parabola is "+ a1+
" x^2 + " +b1 +" y^2 + "+
c1 + " x + " +d1 + " y + "
+ e1+" xy + " + f1 +" = 0.");
}
// Driver Code
public static void Main()
{
float x1 = 0;
float y1 = 0;
float a = 3;
float b = -4;
float c = 2;
equation_parabola(x1, y1, a, b, c);
// This Code is contributed
// by shs
}
}输出：
equation of parabola is 16.0 x^2 + 9.0 y^2 + -12.0 x + 16.0 y + 24.0 xy + -4.0 = 0.
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