Solve conic equation Certainly! Below is a transcription of the image, formatted as it might appear on an educational website:---### Understanding Conic Sections: Circles and EllipsesIn this lesson, we will explore two important conic sections: circles and ellipses. These geometric shapes are defined by specific algebraic equations, which we will analyze and interpret.#### Equation of a CircleThe general equation of a circle with center (h, k) and radius r is given by:\[ (x - h)^2 + (y - k)^2 = r^2 \]In the example provided, we have a circle centered at the origin (0, 0) with a radius squared equal to 13. This is represented by the equation:\[ x^2 + y^2 = 13 \]- **Graph Description:** On the coordinate plane, this circle has its center at the origin and a radius of $\sqrt{13}$.#### Equation of an EllipseThe standard form of an ellipse with center (h, k) is:\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]where $2a$ and $2b$ are the lengths of the major and minor axes, respectively.In the given equation of the ellipse:\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]- **Graph Description:** This ellipse is centered at the origin (0, 0). By comparing to the standard form, we see that $a^2 = 16$ and $b^2 = 4$, giving us: - The length of the semi-major axis $a = 4$ - The length of the semi-minor axis $b = 2$Thus, this ellipse extends 4 units along the x-axis and 2 units along the y-axis from the center.---These equations illustrate the fundamental properties and graphical representations of circles and ellipses, aiding in the understanding of conic sections in algebra and geometry.

The given equations are: From equation (i), we get Substituting this value in (ii), we get

Answered: x? + y? = 13 x2 = 1 16 4

Math

Calculus

x? + y? = 13 x2 = 1 16 4

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Solve conic equation

Certainly! Below is a transcription of the image, formatted as it might appear on an educational website:

---

### Understanding Conic Sections: Circles and Ellipses

In this lesson, we will explore two important conic sections: circles and ellipses. These geometric shapes are defined by specific algebraic equations, which we will analyze and interpret.

#### Equation of a Circle

The general equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]

In the example provided, we have a circle centered at the origin (0, 0) with a radius squared equal to 13. This is represented by the equation:
\[ x^2 + y^2 = 13 \]

- **Graph Description:** On the coordinate plane, this circle has its center at the origin and a radius of $\sqrt{13}$.

#### Equation of an Ellipse

The standard form of an ellipse with center (h, k) is:
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]
where $2a$ and $2b$ are the lengths of the major and minor axes, respectively.

In the given equation of the ellipse:
\[ \frac{x^2}{16} + \frac{y^2}{4} = 1 \]

- **Graph Description:** This ellipse is centered at the origin (0, 0). By comparing to the standard form, we see that $a^2 = 16$ and $b^2 = 4$, giving us:
- The length of the semi-major axis $a = 4$
- The length of the semi-minor axis $b = 2$

Thus, this ellipse extends 4 units along the x-axis and 2 units along the y-axis from the center.

---

These equations illustrate the fundamental properties and graphical representations of circles and ellipses, aiding in the understanding of conic sections in algebra and geometry.

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