Equation of Chord Joining Two Points Having Eccentric Angle

When studying ellipses in the context of conic sections, one of the concepts that arise is the equation of a chord that joins two points on the ellipse. Each point on the ellipse can be represented by its eccentric angle, which is the angle formed by the radius vector with the major axis. In this content, we will explore the equation of such a chord, its derivation, and examples.

Understanding Eccentric Angle

Before we delve into the equation of the chord, let's understand what an eccentric angle is. The eccentric angle, often denoted by $\theta$, is the angle at the center of the ellipse corresponding to a point on the ellipse. It is used to parametrically represent the coordinates of a point on the ellipse.

For an ellipse with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ is the semi-major axis and $b$ is the semi-minor axis, the parametric equations are:

$$ x = a \cos \theta \ y = b \sin \theta $$

Equation of the Chord

Given two points $P_1$ and $P_2$ on the ellipse with eccentric angles $\theta_1$ and $\theta_2$, respectively, the equation of the chord joining these two points can be derived using the parametric coordinates of these points.

Let $P_1(a \cos \theta_1, b \sin \theta_1)$ and $P_2(a \cos \theta_2, b \sin \theta_2)$ be the two points on the ellipse. The slope of the chord joining these points is given by:

$$ m = \frac{b \sin \theta_2 - b \sin \theta_1}{a \cos \theta_2 - a \cos \theta_1} $$

Using the point-slope form of the equation of a line, the equation of the chord is:

$$ y - b \sin \theta_1 = m (x - a \cos \theta_1) $$

Substituting the value of $m$ and simplifying, we get:

$$ \frac{y}{b} - \sin \theta_1 = \frac{\sin \theta_2 - \sin \theta_1}{\cos \theta_2 - \cos \theta_1} \left( \frac{x}{a} - \cos \theta_1 \right) $$

This can be further simplified to the standard form of the equation of the chord joining two points with eccentric angles $\theta_1$ and $\theta_2$.

Differences and Important Points

Here is a table summarizing some important points and differences:

Feature	Description
Eccentric Angle	The angle at the center of the ellipse corresponding to a point on the ellipse.
Parametric Equations	Used to represent the coordinates of a point on the ellipse based on the eccentric angle.
Slope of the Chord	Calculated using the difference in $y$-coordinates over the difference in $x$-coordinates of the two points.
Equation of the Chord	Derived using the point-slope form and the parametric coordinates of the points.

Examples

Let's look at an example to understand the concept better.

Example 1

Consider an ellipse with the equation $\frac{x^2}{16} + \frac{y^2}{9} = 1$. Find the equation of the chord joining the points with eccentric angles $\theta_1 = \frac{\pi}{6}$ and $\theta_2 = \frac{\pi}{3}$.

Solution:

The parametric coordinates of the points are:

$$ P_1(4 \cos \frac{\pi}{6}, 3 \sin \frac{\pi}{6}) = (2\sqrt{3}, \frac{3}{2}) \ P_2(4 \cos \frac{\pi}{3}, 3 \sin \frac{\pi}{3}) = (2, \frac{3\sqrt{3}}{2}) $$

The slope of the chord is:

$$ m = \frac{\frac{3\sqrt{3}}{2} - \frac{3}{2}}{2 - 2\sqrt{3}} = \sqrt{3} $$

Using the point-slope form:

$$ y - \frac{3}{2} = \sqrt{3} (x - 2\sqrt{3}) $$

Simplifying, we get the equation of the chord:

$$ y = \sqrt{3}x - 3 $$

This is the equation of the chord joining the two points on the ellipse with the given eccentric angles.

Example 2

For the same ellipse, find the equation of the chord joining the points with eccentric angles $\theta_1 = 0$ and $\theta_2 = \frac{\pi}{2}$.

Solution:

The parametric coordinates of the points are:

$$ P_1(4 \cos 0, 3 \sin 0) = (4, 0) \ P_2(4 \cos \frac{\pi}{2}, 3 \sin \frac{\pi}{2}) = (0, 3) $$

The slope of the chord is:

$$ m = \frac{3 - 0}{0 - 4} = -\frac{3}{4} $$

Using the point-slope form:

$$ y - 0 = -\frac{3}{4} (x - 4) $$

Simplifying, we get the equation of the chord:

$$ y = -\frac{3}{4}x + 3 $$

This is the equation of the chord joining the two points on the ellipse with the given eccentric angles.

In conclusion, the equation of the chord joining two points with given eccentric angles on an ellipse can be derived using parametric coordinates and the slope of the chord. Understanding this concept is crucial for solving problems related to chords and ellipses in mathematics.