Equation of the Given Ellipse

Oct 13, 2025 by talynt.org 115 views

Introduction

In mathematics, an ellipse is a fundamental concept that has been studied extensively. It is a curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant. 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, respectively. In this article, we will explore the concept of tangent lines to an ellipse and identify the tangent lines to the given ellipse $9x^2 + 16y^2 - 144 = 0$ .

Understanding Ellipses

An ellipse is a closed curve that is symmetrical about its center. The major axis of an ellipse is the longest diameter, while the minor axis is the shortest diameter. The foci of an ellipse are two points inside the ellipse that are equidistant from the center. The sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis.

Equation of the Given Ellipse

The given ellipse is represented by the equation $9x^2 + 16y^2 - 144 = 0$ . To rewrite this equation in standard form, we can divide both sides by 144, which gives us $\frac{9x^2}{144} + \frac{16y^2}{144} = 1$ . Simplifying further, we get $\frac{x^2}{16} + \frac{y^2}{9} = 1$ . Comparing this equation with the standard form of an ellipse, we can see that the center of the ellipse is at the origin $(0,0)$ , and the semi-major and semi-minor axes are $a = 4$ and $b = 3$ , respectively.

Tangent Lines to an Ellipse

A tangent line to an ellipse is a line that touches the ellipse at a single point. The equation of a tangent line to an ellipse can be found using the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$ , where $(x_1,y_1)$ is a point on the ellipse, and $m$ is the slope of the line.

Finding the Slope of the Tangent Line

To find the slope of the tangent line, we can use the fact that the slope of the tangent line to an ellipse at a point $(x_1,y_1)$ is given by the derivative of the equation of the ellipse with respect to $x$ at that point. In this case, the equation of the ellipse is $\frac{x^2}{16} + \frac{y^2}{9} = 1$ . Differentiating both sides with respect to $x$ , we get $\frac{x}{8} + \frac{2y}{9}\frac{dy}{dx} = 0$ . Solving for $\frac{dy}{dx}$ , we get $\frac{dy}{dx} = -\frac{9x}{16y}$ .

Identifying the Tangent Lines

Now that we have the slope of the tangent line, we can use the point-slope form of a line to find the equation of the tangent line. We are given five options for the equation of the tangent line: $x + y + 1 = 0$ , $y - x - 5 = 0$ , $y - x + 5 = 0$ , $2x + y + 1 = 0$ , and $2x - y = 0$ . To identify the correct equation of the tangent line, we can substitute the slope of the tangent line into each of the options and see which one satisfies the equation.

Option A: $x + y + 1 = 0$

Substituting the slope of the tangent line into the equation $x + y + 1 = 0$ , we get $x - \frac{9x}{16y}y + 1 = 0$ . Simplifying further, we get $x - \frac{9x}{16} + 1 = 0$ . Multiplying both sides by 16, we get $16x - 9x + 16 = 0$ . Combining like terms, we get $7x + 16 = 0$ . Subtracting 16 from both sides, we get $7x = -16$ . Dividing both sides by 7, we get $x = -\frac{16}{7}$ . Substituting this value of $x$ into the equation of the ellipse, we get $\frac{\left(-\frac{16}{7}\right)^2}{16} + \frac{y^2}{9} = 1$ . Simplifying further, we get $\frac{256}{784} + \frac{y^2}{9} = 1$ . Multiplying both sides by 784, we get $256 + 88y^2 = 784$ . Subtracting 256 from both sides, we get $88y^2 = 528$ . Dividing both sides by 88, we get $y^2 = 6$ . Taking the square root of both sides, we get $y = \pm\sqrt{6}$ . Substituting these values of $y$ into the equation of the tangent line, we get $x + \pm\sqrt{6} + 1 = 0$ . Simplifying further, we get $x + 1 = 0$ or $x + 1 = 0$ . This is not a valid equation of the tangent line.

Option B: $y - x - 5 = 0$

Substituting the slope of the tangent line into the equation $y - x - 5 = 0$ , we get $-\frac{9x}{16y}y - x - 5 = 0$ . Simplifying further, we get $-\frac{9x}{16} - x - 5 = 0$ . Multiplying both sides by 16, we get $-9x - 16x - 80 = 0$ . Combining like terms, we get $-25x - 80 = 0$ . Adding 80 to both sides, we get $-25x = 80$ . Dividing both sides by -25, we get $x = -\frac{80}{25}$ . Substituting this value of $x$ into the equation of the ellipse, we get $\frac{\left(-\frac{80}{25}\right)^2}{16} + \frac{y^2}{9} = 1$ . Simplifying further, we get $\frac{6400}{1600} + \frac{y^2}{9} = 1$ . Multiplying both sides by 1600, we get $6400 + 1760y^2 = 1600$ . Subtracting 6400 from both sides, we get $1760y^2 = -4800$ . Dividing both sides by 1760, we get $y^2 = -\frac{4800}{1760}$ . This is not a valid equation of the tangent line.

Option C: $y - x + 5 = 0$

Substituting the slope of the tangent line into the equation $y - x + 5 = 0$ , we get $-\frac{9x}{16y}y - x + 5 = 0$ . Simplifying further, we get $-\frac{9x}{16} - x + 5 = 0$ . Multiplying both sides by 16, we get $-9x - 16x + 80 = 0$ . Combining like terms, we get $-25x + 80 = 0$ . Subtracting 80 from both sides, we get $-25x = -80$ . Dividing both sides by -25, we get $x = \frac{80}{25}$ . Substituting this value of $x$ into the equation of the ellipse, we get $\frac{\left(\frac{80}{25}\right)^2}{16} + \frac{y^2}{9} = 1$ . Simplifying further, we get $\frac{6400}{1600} + \frac{y^2}{9} = 1$ . Multiplying both sides by 1600, we get $6400 + 1760y^2 = 1600$ . Subtracting 6400 from both sides, we get $1760y^2 = -4800$ . Dividing both sides by 1760, we get $y^2 = -\frac{4800}{1760}$ . This is not a valid equation of the tangent line.

Option D: $2x + y + 1 = 0$

Q: What is an ellipse?

A: An ellipse is a fundamental concept in mathematics that is a closed curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant.

Q: What is the equation of an ellipse?

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, respectively.

Q: What is a tangent line to an ellipse?

A: A tangent line to an ellipse is a line that touches the ellipse at a single point.

Q: How do you find the slope of the tangent line to an ellipse?

A: To find the slope of the tangent line to an ellipse, you can use the fact that the slope of the tangent line to an ellipse at a point $(x_1,y_1)$ is given by the derivative of the equation of the ellipse with respect to $x$ at that point.

Q: What is the equation of the tangent line to the given ellipse?

A: The equation of the tangent line to the given ellipse is $2x - y = 0$ .

Q: How do you identify the tangent line to an ellipse?

A: To identify the tangent line to an ellipse, you can substitute the slope of the tangent line into each of the options and see which one satisfies the equation.

Q: What are the characteristics of the tangent line to an ellipse?

A: The tangent line to an ellipse is a line that touches the ellipse at a single point and has a slope that is equal to the derivative of the equation of the ellipse with respect to $x$ at that point.

Q: Can you provide an example of finding the tangent line to an ellipse?

A: Yes, let's consider the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ . To find the tangent line to this ellipse, we can use the fact that the slope of the tangent line to an ellipse at a point $(x_1,y_1)$ is given by the derivative of the equation of the ellipse with respect to $x$ at that point. In this case, the derivative of the equation of the ellipse with respect to $x$ is $\frac{dy}{dx} = -\frac{9x}{16y}$ . Substituting this value of $\frac{dy}{dx}$ into the equation of the tangent line, we get $y - x - 5 = 0$ . Simplifying further, we get $y - x + 5 = 0$ . This is the equation of the tangent line to the given ellipse.

Q: What are the applications of tangent lines to ellipses?

A: Tangent lines to ellipses have numerous applications in mathematics, physics, and engineering. Some of the applications include:

Optics: Tangent lines to ellipses are used to describe the path of light rays as they pass through an optical system.
Astronomy: Tangent lines to ellipses are used to describe the orbits of celestial bodies, such as planets and comets.
Engineering: Tangent lines to ellipses are used to design and optimize systems, such as gears and pulleys.

Q: Can you provide a summary of the key concepts related to tangent lines to ellipses?

A: Yes, here is a summary of the key concepts related to tangent lines to ellipses:

Definition: A tangent line to an ellipse is a line that touches the ellipse at a single point.
Equation: The equation of a tangent line to an ellipse is given by $y - y_1 = m(x - x_1)$ , where $(x_1,y_1)$ is a point on the ellipse, and $m$ is the slope of the line.
Slope: The slope of the tangent line to an ellipse at a point $(x_1,y_1)$ is given by the derivative of the equation of the ellipse with respect to $x$ at that point.
Characteristics: The tangent line to an ellipse is a line that touches the ellipse at a single point and has a slope that is equal to the derivative of the equation of the ellipse with respect to $x$ at that point.