**Which of the following is not a parabola equation?**

** ****Options:**

A. x2 = 4ay B. y2 – 8ax = 0 C. x2 = by D. x2 = 4ay2 |

### The Correct Answer Is:

- D. x2 = 4ay2

The correct answer is D) x^2 = 4ay^2.

**Understanding Parabola Equations:**

A parabola is a fundamental geometric shape that is defined as a set of points that are equidistant from a fixed point (the focus) and a fixed straight line (the directrix). Parabolas come in two main orientations: vertical and horizontal, and their equations have distinct forms based on this orientation.

In the context of the standard form of a parabola equation, there are two main categories:

**Vertical Parabolas:**

These are parabolas that open either upward or downward and have equations in which the squared variable (usually y) is proportional to the other variable (x).

**Horizontal Parabolas:**

These are parabolas that open either to the right or to the left and have equations in which the squared variable (usually x) is proportional to the other variable (y).

Now, let’s examine why “x^2 = 4ay^2” does not fit the standard form of a parabola equation and why it is not a valid parabola equation.

**Why “x^2 = 4ay^2” is Not a Parabola Equation:**

**1. Lack of Proportionality:**

In a standard parabola equation, one variable is squared and directly proportional to the other variable. In the equation “y^2 = 4ax,” for example, the squared variable (y^2) is directly proportional to x.

This proportionality is what defines the shape and characteristics of a parabola. In “x^2 = 4ay^2,” both x^2 and y^2 are squared, but they are not directly proportional to each other. This lack of proportionality means that the equation does not represent the standard form of a parabola.

**2. Symmetry and Orientation:**

The orientation of a parabola is crucial in determining its direction of opening. Vertical parabolas open either upward or downward, while horizontal parabolas open either to the right or to the left.

In the equation “y^2 = 4ax,” the squared variable (y^2) indicates that it is a vertical parabola opening either upward or downward, depending on the sign of “a.” In “x^2 = 4ay^2,” both x^2 and y^2 are squared, which does not conform to the typical orientation of a parabola. This equation lacks the characteristics of a standard vertical or horizontal parabola.

**3. General Characteristics:**

Parabolas have specific characteristics, including a focus (or focal point), a directrix (or directrix line), and a vertex. These characteristics define the shape and position of the parabola. The coefficients in the standard form of a parabola equation determine these characteristics.

In the equation “x^2 = 4ay^2,” there are no coefficients that define the focus, directrix, or vertex. This absence of coefficients related to these characteristics makes it incompatible with the standard properties of parabolas.

**Why the Other Options Are Correct Parabola Equations:**

**A. x^2 = 4ay:**

This equation represents a horizontal parabola that opens to the right. It conforms to the standard form of a parabola equation for horizontal parabolas. The squared variable (x^2) is proportional to y, and the coefficient “4a” determines the focus and directrix characteristics. This equation accurately describes a valid horizontal parabola.

**B. y^2 – 8ax = 0:**

This equation represents a vertical parabola that opens upward. It conforms to the standard form of a parabola equation for vertical parabolas. The squared variable (y^2) is proportional to x, and the coefficient “8a” determines the focus and directrix characteristics. This equation accurately describes a valid vertical parabola.

**C. x^2 = by:**

This equation represents a vertical parabola that opens upward. It also conforms to the standard form of a parabola equation for vertical parabolas. The squared variable (x^2) is proportional to y, and the coefficient “b” determines the focus and directrix characteristics. This equation accurately describes a valid vertical parabola.

In summary, “x^2 = 4ay^2” is not a valid parabola equation because it does not follow the standard form, lacks proportionality between the squared variables, and does not adhere to the typical characteristics and orientation of parabolas.

On the other hand, options A, B, and C represent valid parabola equations that conform to the standard forms for vertical and horizontal parabolas, with coefficients that determine the focus, directrix, and shape of the parabolas.

Understanding the properties and equations of parabolas is essential in mathematics and physics, as parabolas appear in various real-world applications, including projectile motion, optics, and engineering. Correctly identifying and working with parabola equations helps in solving mathematical and physical problems with precision and accuracy.

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