Chord of contact
Chord of Contact
The chord of contact refers to a specific line segment within the context of conic sections such as ellipses, parabolas, and hyperbolas. When a point external to the conic section is taken, and tangents are drawn from this point to the conic, the chord of contact is the line segment that connects the points of tangency on the conic.
Understanding Chord of Contact in Ellipses
For an ellipse, the chord of contact is particularly interesting because it has a unique property: it is the locus of the points of intersection of pairs of tangents to the ellipse, drawn from points outside the ellipse.
Equation of the Chord of Contact
Given 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 equation of the chord of contact for a point $(x_1, y_1)$ outside the ellipse is given by:
$$ \frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1 $$
This equation is derived from the equation of the tangent to the ellipse at any point $(x, y)$ on it.
Examples
Let's consider an ellipse with the equation $\frac{x^2}{25} + \frac{y^2}{16} = 1$. For a point $(3, 4)$ outside the ellipse, the equation of the chord of contact would be:
$$ \frac{x \cdot 3}{25} + \frac{y \cdot 4}{16} = 1 $$
Simplifying, we get:
$$ \frac{3x}{25} + \frac{y}{4} = 1 $$
Important Points and Differences
| Feature | Description |
|---|---|
| Definition | The chord of contact is the line segment that connects the points of tangency of tangents drawn from an external point to the ellipse. |
| Equation | For an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the chord of contact from a point $(x_1, y_1)$ is $\frac{x x_1}{a^2} + \frac{y y_1}{b^2} = 1$. |
| Locus Property | The chord of contact is the locus of the points of intersection of pairs of tangents to the ellipse, drawn from points outside the ellipse. |
| Relation to Tangents | The chord of contact is directly related to the tangents of the ellipse. It is the line that would be formed if the tangents from an external point were to be extended until they meet the ellipse. |
Practice Problems
Find the equation of the chord of contact of the tangents drawn from the point $(4, 5)$ to the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$.
If the chord of contact of the tangents drawn from a point P to the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ passes through the point $(1, 2)$, find the coordinates of point P.
By understanding the concept of the chord of contact and its equation, students can solve various problems related to tangents and normals of ellipses, which are common in exams involving conic sections.