A Parabola Can Be Represented By The Equation X − 2 = − 20 Y X^{-2} = -20y X − 2 = − 20 Y . Which Of The Following Points Lies On The Parabola?A. (-5, 0) B. (5, 0) C. (0, 5) D. (0, -5)

Introduction to Parabolas

A parabola is a fundamental concept in mathematics, representing a U-shaped curve. It can be defined as the set of all points that are equidistant from a fixed point (known as the focus) and a fixed line (known as the directrix). Parabolas have numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on identifying points that lie on a given parabola represented by the equation $x^{-2} = -20y$.

The Equation of a Parabola

The given equation $x^{-2} = -20y$ represents a parabola in the Cartesian coordinate system. To understand this equation, let's break it down into its components. The term $x^{-2}$ represents the reciprocal of $x^2$, which means that as $x$ increases, the value of $x^{-2}$ decreases, and vice versa. The term $-20y$ represents a linear function that is multiplied by $-20$. The negative sign indicates that the parabola opens downwards.

Identifying Points on the Parabola

To identify points that lie on the parabola, we need to substitute the coordinates of each point into the equation and check if the equation holds true. Let's analyze each option:

Option A: (-5, 0)

Substituting the coordinates (-5, 0) into the equation, we get:

$$(-5)^{-2} = -20(0)$$

Simplifying the equation, we get:

$$\frac{1}{25} = 0$$

This equation is not true, as $\frac{1}{25}$ is not equal to 0. Therefore, point A (-5, 0) does not lie on the parabola.

Option B: (5, 0)

Substituting the coordinates (5, 0) into the equation, we get:

$$(5)^{-2} = -20(0)$$

Simplifying the equation, we get:

$$\frac{1}{25} = 0$$

This equation is not true, as $\frac{1}{25}$ is not equal to 0. Therefore, point B (5, 0) does not lie on the parabola.

Option C: (0, 5)

Substituting the coordinates (0, 5) into the equation, we get:

$$(0)^{-2} = -20(5)$$

Simplifying the equation, we get:

$$\text{undefined} = -100$$

This equation is not true, as the left-hand side is undefined (division by zero). Therefore, point C (0, 5) does not lie on the parabola.

Option D: (0, -5)

Substituting the coordinates (0, -5) into the equation, we get:

$$(0)^{-2} = -20(-5)$$

Simplifying the equation, we get:

$$\text{undefined} = 100$$

This equation is not true, as the left-hand side is undefined (division by zero). However, we can see that the right-hand side is a positive value, which means that the point (0, -5) is not on the parabola.

Conclusion

After analyzing each option, we can conclude that none of the points A, B, C, or D lie on the parabola represented by the equation $x^{-2} = -20y$. This is because each point either does not satisfy the equation or is undefined due to division by zero.

Understanding the Limitations of the Equation

The equation $x^{-2} = -20y$ has a limitation in that it is undefined when $x = 0$. This is because division by zero is undefined in mathematics. Therefore, any point that has an x-coordinate of 0 will not lie on the parabola.

Real-World Applications of Parabolas

Parabolas have numerous real-world applications, including:

• Physics: Parabolas are used to describe the trajectory of projectiles under the influence of gravity.
• Engineering: Parabolas are used in the design of antennas, satellite dishes, and other communication systems.
• Computer Science: Parabolas are used in computer graphics to create smooth curves and shapes.

Conclusion

In conclusion, the equation $x^{-2} = -20y$ represents a parabola in the Cartesian coordinate system. By analyzing each option, we can conclude that none of the points A, B, C, or D lie on the parabola. This is because each point either does not satisfy the equation or is undefined due to division by zero. Understanding the limitations of the equation and its real-world applications can help us appreciate the importance of parabolas in mathematics and beyond.

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Introduction

In our previous article, we explored the concept of parabolas and identified points that lie on a given parabola represented by the equation $x^{-2} = -20y$. In this article, we will answer some frequently asked questions (FAQs) related to parabolas and the given equation.

Q: What is a parabola?

A: A parabola is a U-shaped curve that can be defined as the set of all points that are equidistant from a fixed point (known as the focus) and a fixed line (known as the directrix).

Q: What is the equation $x^{-2} = -20y$?

A: The equation $x^{-2} = -20y$ represents a parabola in the Cartesian coordinate system. The term $x^{-2}$ represents the reciprocal of $x^2$, and the term $-20y$ represents a linear function that is multiplied by $-20$.

Q: Why is the equation $x^{-2} = -20y$ undefined when $x = 0$?

A: The equation $x^{-2} = -20y$ is undefined when $x = 0$ because division by zero is undefined in mathematics. When $x = 0$, the term $x^{-2}$ becomes undefined, and the equation cannot be solved.

Q: Can you provide an example of a point that lies on the parabola represented by the equation $x^{-2} = -20y$?

A: Unfortunately, we were unable to find any points that lie on the parabola represented by the equation $x^{-2} = -20y$. However, we can provide an example of a point that lies on a different parabola. For example, the point (2, -1) lies on the parabola represented by the equation $y = x^2$.

Q: What are some real-world applications of parabolas?

A: Parabolas have numerous real-world applications, including:

• Physics: Parabolas are used to describe the trajectory of projectiles under the influence of gravity.
• Engineering: Parabolas are used in the design of antennas, satellite dishes, and other communication systems.
• Computer Science: Parabolas are used in computer graphics to create smooth curves and shapes.

Q: How can I graph a parabola represented by an equation?

A: To graph a parabola represented by an equation, you can use a graphing calculator or a computer program. You can also use a piece of graph paper and a pencil to plot points on the parabola.

Q: Can you provide a table of values for the parabola represented by the equation $x^{-2} = -20y$?

A: Unfortunately, we were unable to provide a table of values for the parabola represented by the equation $x^{-2} = -20y$ because the equation is undefined when $x = 0$. However, we can provide a table of values for a different parabola. For example, the table of values for the parabola represented by the equation $y = x^2$ is:

x	y
-2	4
-1	1
0	0
1	1
2	4

Conclusion

In conclusion, the equation $x^{-2} = -20y$ represents a parabola in the Cartesian coordinate system. By answering some frequently asked questions (FAQs) related to parabolas and the given equation, we can gain a better understanding of the concept of parabolas and their applications in mathematics and beyond.

Additional Resources

For more information on parabolas and their applications, please refer to the following resources:

• Mathematics textbooks: Many mathematics textbooks cover the concept of parabolas and their equations.
• Online resources: Websites such as Khan Academy, Mathway, and Wolfram Alpha provide interactive lessons and examples on parabolas and their equations.
• Graphing calculators: Graphing calculators such as the TI-83 and TI-84 can be used to graph parabolas and other curves.

Final Thoughts

In conclusion, the equation $x^{-2} = -20y$ represents a parabola in the Cartesian coordinate system. By understanding the concept of parabolas and their equations, we can gain a better appreciation for the beauty and complexity of mathematics.