Which Conic Section Is Defined By The Equation Shown Below?$x^2 + Y^2 - 10x - 2y + 10 = 0$A. Circle B. Ellipse C. Parabola D. Hyperbola

Introduction

Conic sections are a fundamental concept in mathematics, and understanding which conic section is defined by a given equation is crucial for solving various mathematical problems. In this article, we will explore the equation $x^2 + y^2 - 10x - 2y + 10 = 0$ and determine which conic section it represents.

What are Conic Sections?

Conic sections are a set of curves that are obtained by intersecting a cone with a plane. The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas. Each of these conic sections has its unique properties and characteristics.

Equation of a Conic Section

The general equation of a conic section is given by:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

where $A$, $B$, $C$, $D$, $E$, and $F$ are constants. The values of these constants determine the type of conic section represented by the equation.

Completing the Square

To identify the conic section represented by the equation $x^2 + y^2 - 10x - 2y + 10 = 0$, we need to complete the square. This involves rearranging the equation to express it in the form:

$$(x - h)^2 + (y - k)^2 = r^2$$

where $(h, k)$ is the center of the conic section and $r$ is the radius.

Completing the Square for the Given Equation

Let's complete the square for the given equation:

$$x^2 + y^2 - 10x - 2y + 10 = 0$$

First, group the $x$-terms and $y$-terms:

$$x^2 - 10x + y^2 - 2y = -10$$

Next, add and subtract the square of half the coefficient of $x$ and $y$:

$$x^2 - 10x + 25 - 25 + y^2 - 2y + 1 - 1 = -10$$

Now, factor the perfect square trinomials:

$$(x - 5)^2 - 25 + (y - 1)^2 - 1 = -10$$

Combine like terms:

$$(x - 5)^2 + (y - 1)^2 = 16$$

Identifying the Conic Section

The equation $(x - 5)^2 + (y - 1)^2 = 16$ represents a circle with center $(5, 1)$ and radius $4$. Therefore, the conic section defined by the equation $x^2 + y^2 - 10x - 2y + 10 = 0$ is a circle.

Conclusion

In this article, we have identified the conic section represented by the equation $x^2 + y^2 - 10x - 2y + 10 = 0$. By completing the square, we were able to express the equation in the form of a circle. This demonstrates the importance of completing the square in identifying conic sections.

Key Takeaways

• Conic sections are a set of curves obtained by intersecting a cone with a plane.
• The general equation of a conic section is given by $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
• Completing the square is a crucial step in identifying the conic section represented by an equation.
• The equation $x^2 + y^2 - 10x - 2y + 10 = 0$ represents a circle with center $(5, 1)$ and radius $4$.

Frequently Asked Questions

Q: What is the general equation of a conic section?

A: The general equation of a conic section is given by $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

Q: How do I complete the square for a given equation?

A: To complete the square, group the $x$-terms and $y$-terms, add and subtract the square of half the coefficient of $x$ and $y$, and factor the perfect square trinomials.

Q: What is the center and radius of the circle represented by the equation $x^2 + y^2 - 10x - 2y + 10 = 0$?

A: The center of the circle is $(5, 1)$ and the radius is $4$.

Q: What are the four basic types of conic sections?

Introduction

Conic sections are a fundamental concept in mathematics, and understanding the different types of conic sections, their equations, and properties is crucial for solving various mathematical problems. In this article, we will provide a comprehensive Q&A guide on conic sections, covering topics such as the general equation of a conic section, completing the square, and identifying the type of conic section represented by a given equation.

Q&A: Conic Sections

Q: What is the general equation of a conic section?

A: The general equation of a conic section is given by $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where $A$, $B$, $C$, $D$, $E$, and $F$ are constants.

Q: What are the four basic types of conic sections?

A: The four basic types of conic sections are circles, ellipses, parabolas, and hyperbolas.

Q: How do I identify the type of conic section represented by a given equation?

A: To identify the type of conic section, you need to complete the square and express the equation in the form of a circle, ellipse, parabola, or hyperbola.

Q: What is completing the square?

A: Completing the square is a process of rearranging the equation to express it in the form of a perfect square trinomial, which can be factored into the square of a binomial.

Q: How do I complete the square for a given equation?

A: To complete the square, group the $x$-terms and $y$-terms, add and subtract the square of half the coefficient of $x$ and $y$, and factor the perfect square trinomials.

Q: What is the center and radius of the circle represented by the equation $x^2 + y^2 - 10x - 2y + 10 = 0$?

A: The center of the circle is $(5, 1)$ and the radius is $4$.

Q: What is the equation of a circle in standard form?

A: The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is the radius.

Q: What is the equation of an ellipse in standard form?

A: The equation of an ellipse in standard form is given by $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where $(h, k)$ is the center of the ellipse and $a$ and $b$ are the semi-major and semi-minor axes.

Q: What is the equation of a parabola in standard form?

A: The equation of a parabola in standard form is given by $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the equation of a hyperbola in standard form?

A: The equation of a hyperbola in standard form is given by $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$, where $(h, k)$ is the center of the hyperbola and $a$ and $b$ are the distances from the center to the vertices.

Conclusion

In this article, we have provided a comprehensive Q&A guide on conic sections, covering topics such as the general equation of a conic section, completing the square, and identifying the type of conic section represented by a given equation. We hope that this guide has been helpful in understanding the different types of conic sections and their properties.

Key Takeaways

• The general equation of a conic section is given by $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
• Completing the square is a process of rearranging the equation to express it in the form of a perfect square trinomial.
• The equation of a circle in standard form is given by $(x - h)^2 + (y - k)^2 = r^2$.
• The equation of an ellipse in standard form is given by $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$.
• The equation of a parabola in standard form is given by $y = ax^2 + bx + c$.
• The equation of a hyperbola in standard form is given by $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$.

Frequently Asked Questions

Q: What is the difference between a circle and an ellipse?

A: A circle is a set of points that are equidistant from a central point, while an ellipse is a set of points that are equidistant from two central points.

Q: What is the difference between a parabola and a hyperbola?

A: A parabola is a set of points that are equidistant from a central point and a line, while a hyperbola is a set of points that are equidistant from two central points and a line.

Q: How do I graph a conic section?

A: To graph a conic section, you need to identify the center, vertices, and axes of the conic section, and then plot the points on a coordinate plane.

Q: What is the importance of conic sections in real-life applications?

A: Conic sections are used in various real-life applications, such as architecture, engineering, and physics, to model and analyze the behavior of objects and systems.