Understanding the Parabola and Its Directrix

Parabolas are a fundamental concept in mathematics, arising in various fields such as physics, engineering, and geometry. A parabola is the set of points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix). In this article, we will explore how to find the equation of the directrix for a given parabola, using the example y^2 4x -3.

The Parabola Equation

The given parabola equation is y^2 4x - 3. To find the equation of the directrix, we need to first rewrite the equation in the standard form. The standard form for a parabola is y^2 4px, where p is the focal length, and the vertex is at the origin (0, 0).

Rewriting the Equation in Standard Form

Starting with the given equation y^2 4x - 3, we can rewrite it as: [y^2 4(x - frac{3}{4})]

By comparing this with the standard form y^2 4px, we can identify the parameters:

- The vertex of the parabola is at (h, k) left(-frac{3}{4}, 0right). - The coefficient of x is -4, so we can solve for p as follows: 4p -4, which gives p -1.

Equation of the Directrix

The directrix of a parabola that opens to the left or to the right is given by x h - p. Since we have identified the vertex (h, k) left(-frac{3}{4}, 0right) and p -1, we can now find the equation of the directrix:

[x -frac{3}{4} - (-1) -frac{3}{4} 1 frac{1}{4}]

Therefore, the equation of the directrix is x frac{1}{4}.

Conclusion and Generalization

This example demonstrates the process of transforming a given parabola equation into its standard form and then applying the formula for the directrix to find its equation. The key steps are:

1. Rewrite the given equation in the form y^2 4px (if not already in this form). 2. Identify the vertex and the value of p from the equation. 3. Substitute the vertex and p into the directrix formula x h - p to find the equation of the directrix.

By following these steps, you can find the equation of the directrix for any parabola, regardless of its orientation or position on the coordinate plane.

Key Takeaways

- The standard form of a rightwards-opening parabola is y^2 4px. - The equation of the directrix is x h - p, where (h, k) is the vertex and p is the focal length. Related Keywords - Parabola - Directrix - Vertex - Standard Form - Focus - Focal Length

Qwen provided by Alibaba Cloud