Find the slope of the tangent line

Oct 13, 2025 by leolavanderia.com.br 106 views

Introduction

In mathematics, the study of tangent lines to curves is a fundamental concept that has numerous applications in various fields, including physics, engineering, and computer science. An ellipse is a classic example of a curve that can have multiple tangent lines, each with its own unique characteristics. In this article, we will delve into the world of ellipses and explore the process of identifying tangent lines to a given ellipse. We will also examine the tangent points of each line and provide a comprehensive understanding of the underlying mathematics.

Understanding Ellipses

An ellipse is a closed curve in a plane that results from the intersection of a cone and a plane cutting the cone at an angle. It is characterized by its major and minor axes, which are the longest and shortest diameters of the ellipse, respectively. The general equation of an ellipse is given by:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

The Given Ellipse

The given ellipse is represented by the equation:

9x^2 + 16y^2 - 144 = 0

To rewrite this equation in the standard form, we can divide both sides by 144:

\frac{9x^2}{144} + \frac{16y^2}{144} = 1

Simplifying further, we get:

\frac{x^2}{16} + \frac{y^2}{9} = 1

This equation represents an ellipse with semi-major axis $a = 4$ and semi-minor axis $b = 3$ .

Identifying Tangent Lines

To identify the tangent lines to the ellipse, we need to find the lines that are tangent to the ellipse at a point. A tangent line is a line that touches the ellipse at exactly one point. To find the tangent lines, we can use the following approach:

Find the slope of the tangent line: The slope of the tangent line at a point $(x_0, y_0)$ on the ellipse is given by the derivative of the equation of the ellipse at that point.
Find the equation of the tangent line: Using the slope of the tangent line and the point of tangency, we can find the equation of the tangent line.

Tangent Line (a)

The first tangent line is given by the equation:

x + y + 1 = 0

To find the slope of this line, we can rewrite it in the form $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept:

y = -x - 1

The slope of this line is $m = -1$ .

Finding the Point of Tangency

To find the point of tangency, we need to substitute the equation of the line into the equation of the ellipse:

\frac{x^2}{16} + \frac{(-x - 1)^2}{9} = 1

Simplifying, we get:

\frac{x^2}{16} + \frac{x^2 + 2x + 1}{9} = 1

Combine like terms:

\frac{9x^2 + 16x^2 + 32x + 16}{144} = 1

Simplify further:

\frac{25x^2 + 32x + 16}{144} = 1

Multiply both sides by 144:

25x^2 + 32x + 16 = 144

Subtract 144 from both sides:

25x^2 + 32x - 128 = 0

Divide both sides by 4:

6.25x^2 + 8x - 32 = 0

Solve for x using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a = 6.25$ , $b = 8$ , and $c = -32$ .

x = \frac{-8 \pm \sqrt{8^2 - 4(6.25)(-32)}}{2(6.25)}

Simplify:

x = \frac{-8 \pm \sqrt{64 + 800}}{12.5}

x = \frac{-8 \pm \sqrt{864}}{12.5}

x = \frac{-8 \pm 29.39}{12.5}

Solve for x:

x = \frac{-8 + 29.39}{12.5}

x = \frac{21.39}{12.5}

x = 1.71

x = \frac{-8 - 29.39}{12.5}

x = \frac{-37.39}{12.5}

x = -3

Substitute x into the equation of the line to find y:

y = -x - 1

y = -1.71 - 1

y = -2.71

y = -(-3) - 1

y = 2

The point of tangency is $(1.71, -2.71)$ .

Tangent Line (b)

The second tangent line is given by the equation:

y - x - 5 = 0

To find the slope of this line, we can rewrite it in the form $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept:

y = x + 5

The slope of this line is $m = 1$ .

Finding the Point of Tangency

To find the point of tangency, we need to substitute the equation of the line into the equation of the ellipse:

\frac{x^2}{16} + \frac{(x + 5)^2}{9} = 1

Simplifying, we get:

\frac{x^2}{16} + \frac{x^2 + 10x + 25}{9} = 1

Combine like terms:

\frac{9x^2 + 16x^2 + 160x + 400}{144} = 1

Simplify further:

\frac{25x^2 + 160x + 400}{144} = 1

Multiply both sides by 144:

25x^2 + 160x + 400 = 144

Subtract 144 from both sides:

25x^2 + 160x + 256 = 0

Divide both sides by 4:

6.25x^2 + 40x + 64 = 0

Solve for x using the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where $a = 6.25$ , $b = 40$ , and $c = 64$ .

x = \frac{-40 \pm \sqrt{40^2 - 4(6.25)(64)}}{2(6.25)}

Simplify:

x = \frac{-40 \pm \sqrt{1600 - 1600}}{12.5}

x = \frac{-40 \pm \sqrt{0}}{12.5}

x = \frac{-40}{12.5}

x = -3.2

Substitute x into the equation of the line to find y:

y = x + 5

y = -3.2 + 5

y = 1.8

The point of tangency is $(-3.2, 1.8)$ .

Tangent Line (c)

The third tangent line is given by the equation:

y - x + 5 = 0

To find the slope of this line, we can rewrite it in the form $y = mx + c$ , where $m$ is the slope and $c$ is the y-intercept:

y = x - 5

The slope of this line is $m = 1$ .

Finding the Point of Tangency

To find the point of tangency, we need to substitute the equation of the line into the equation of the ellipse:

\frac{x^2}{16} + \frac{(x - 5)^2}{9} = 1

Simplifying, we get:

\frac{x^2}{16} + \frac{x^2 - 10x + 25}{9} = 1

Combine like terms:

\frac{9x^2 + 16x^2 - 160x + 400}{144} = 1

Simplify further:

\frac{25x^2 - 160x + 400}{144} = 1

Multiply both sides by 144:

25x^2 - 160x + 400 = 144

Subtract 144 from both sides:

25x^2 - 160x + 256 = 0

Divide both sides by 4:

6.25x^2 - 40<br/> # **Frequently Asked Questions: Identifying Tangent Lines to an Ellipse**

Q: What is the significance of identifying tangent lines to an ellipse?

A: Identifying tangent lines to an ellipse is a fundamental concept in mathematics that has numerous applications in various fields, including physics, engineering, and computer science. Tangent lines are used to model real-world problems, such as the motion of objects, the behavior of electrical circuits, and the design of mechanical systems.

Q: How do I find the tangent lines to an ellipse?

A: To find the tangent lines to an ellipse, you need to follow these steps:

Find the equation of the ellipse: The equation of the ellipse is given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.
Find the slope of the tangent line: The slope of the tangent line at a point $(x_0, y_0)$ on the ellipse is given by the derivative of the equation of the ellipse at that point.
Find the equation of the tangent line: Using the slope of the tangent line and the point of tangency, you can find the equation of the tangent line.

Q: What is the point of tangency?

A: The point of tangency is the point on the ellipse where the tangent line touches the ellipse. It is the point where the tangent line is perpendicular to the ellipse.

Q: How do I find the point of tangency?

A: To find the point of tangency, you need to substitute the equation of the line into the equation of the ellipse and solve for the point of intersection.

Q: What are the different types of tangent lines?

A: There are two types of tangent lines:

Tangent line: A tangent line is a line that touches the ellipse at exactly one point.
Secant line: A secant line is a line that intersects the ellipse at two points.

Q: How do I determine if a line is a tangent line or a secant line?

A: To determine if a line is a tangent line or a secant line, you need to check if the line intersects the ellipse at one or two points. If the line intersects the ellipse at one point, it is a tangent line. If the line intersects the ellipse at two points, it is a secant line.

Q: What are the applications of tangent lines in real-world problems?

A: Tangent lines have numerous applications in real-world problems, including:

Physics: Tangent lines are used to model the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
Engineering: Tangent lines are used to design mechanical systems, such as gears and pulleys.
Computer Science: Tangent lines are used in computer graphics to create smooth curves and surfaces.

Q: How do I use tangent lines in computer graphics?

A: To use tangent lines in computer graphics, you need to follow these steps:

Create a curve: Create a curve using a set of points.
Find the tangent line: Find the tangent line at each point on the curve.
Use the tangent line: Use the tangent line to create a smooth curve or surface.

Q: What are the benefits of using tangent lines in computer graphics?

A: The benefits of using tangent lines in computer graphics include:

Smooth curves: Tangent lines create smooth curves and surfaces.
Realistic rendering: Tangent lines create realistic rendering of 3D objects.
Improved performance: Tangent lines improve the performance of computer graphics applications.

Q: How do I implement tangent lines in a programming language?

A: To implement tangent lines in a programming language, you need to follow these steps:

Choose a programming language: Choose a programming language, such as C++ or Python.
Use a library: Use a library, such as OpenCV or NumPy, to implement tangent lines.
Write code: Write code to implement tangent lines.

Q: What are the challenges of implementing tangent lines in a programming language?

A: The challenges of implementing tangent lines in a programming language include:

Complexity: Tangent lines are a complex mathematical concept.
Numerical instability: Tangent lines can be numerically unstable.
Performance: Tangent lines can be computationally expensive.