C/C++ Program for Finding the vertex, focus and directrix of a parabola?

A parabola is a U-shaped curve that can be described by a quadratic equation. In C programming, we can find key properties of a parabola including its vertex, focus, and directrix using mathematical formulas. The general equation of a parabola is −

Syntax

y = ax² + bx + c

Where a, b, and c are coefficients and a ? 0.

Formulas

For a parabola y = ax² + bx + c, the key properties are calculated as −

• Vertex: (-b/(2a), (4ac - b²)/(4a))
• Focus: (-b/(2a), (4ac - b² + 1)/(4a))
• Directrix: y = (4ac - b² - 1)/(4a)

Example

This program calculates the vertex, focus, and directrix of a parabola given its coefficients −

#include <stdio.h>

void getParabolaDetails(float a, float b, float c) {
    float vertex_x = -b / (2 * a);
    float vertex_y = ((4 * a * c) - (b * b)) / (4 * a);
    
    float focus_x = vertex_x;
    float focus_y = vertex_y + (1 / (4 * a));
    
    float directrix = vertex_y - (1 / (4 * a));
    
    printf("Parabola equation: y = %.1fx² + %.1fx + %.1f\n", a, b, c);
    printf("Vertex: (%.3f, %.3f)\n", vertex_x, vertex_y);
    printf("Focus: (%.3f, %.3f)\n", focus_x, focus_y);
    printf("Directrix: y = %.3f\n", directrix);
}

int main() {
    float a = 1, b = -4, c = 3;
    
    printf("Finding parabola properties:\n");
    getParabolaDetails(a, b, c);
    
    return 0;
}

Finding parabola properties:
Parabola equation: y = 1.0x² + -4.0x + 3.0
Vertex: (2.000, -1.000)
Focus: (2.000, -0.750)
Directrix: y = -1.250

Key Points

• The vertex is the lowest (or highest) point of the parabola
• The focus is a point inside the parabola used for its geometric definition
• The directrix is a horizontal line that, together with the focus, defines the parabola
• For parabolas opening upward (a > 0), the focus is above the vertex

Conclusion

Using the standard quadratic equation coefficients, we can easily calculate the vertex, focus, and directrix of a parabola in C. These properties help visualize and analyze the parabola's geometric characteristics.