Chord of contact
Chord of Contact
The concept of the chord of contact is an important topic in the geometry of circles and is particularly relevant in coordinate geometry. It is related to tangents drawn from an external point to a circle.
Definition
When two tangents are drawn to a circle from an external point, they touch the circle at distinct points. The line segment that joins these two points of contact is called the chord of contact.
Equation of the Chord of Contact
For a circle with the equation $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center of the circle and $r$ is its radius, the equation of the chord of contact for tangents drawn from an external point $(x_1, y_1)$ can be derived using the concept of the polar of a point with respect to a circle.
The equation of the chord of contact is given by:
$$ T = 0 $$
where $T$ is the expression obtained by replacing $x^2$ by $xx_1$, $y^2$ by $yy_1$, and the constant term by $x_1x + y_1y$ in the given circle's equation.
For the circle $(x - h)^2 + (y - k)^2 = r^2$, the equation of the chord of contact from the point $(x_1, y_1)$ is:
$$ (x - h)(x_1 - h) + (y - k)(y_1 - k) = r^2 $$
Examples
Let's consider a circle with the equation $x^2 + y^2 = 25$ and an external point $(4, 5)$. The equation of the chord of contact from this point is obtained by replacing $x^2$ with $4x$, $y^2$ with $5y$, and the constant term with $4x + 5y$:
$$ 4x + 5y = 25 $$
This is the equation of the chord of contact from the point $(4, 5)$ to the circle $x^2 + y^2 = 25$.
Table of Differences and Important Points
| Feature | Chord of Contact | Regular Chord |
|---|---|---|
| Definition | Line segment joining points where tangents from an external point touch the circle. | Line segment whose endpoints both lie on the circle. |
| Equation | Derived using the polar of a point with respect to the circle. | Can be found using the midpoint and the perpendicular distance from the center. |
| Relation to External Point | Always related to an external point from which tangents are drawn. | Does not necessarily relate to an external point. |
| Tangency | Formed by the tangency points of two tangents from an external point. | Not necessarily related to tangents. |
Important Points to Remember
- The chord of contact is specific to an external point and the circle in question.
- The length of the chord of contact can be calculated if the radius of the circle and the distance of the external point from the center are known.
- The chord of contact is a special case of the polar of a point with respect to a circle.
Conclusion
The chord of contact is a fundamental concept in circle geometry, especially when dealing with tangents from an external point. Understanding how to derive and use the equation of the chord of contact is essential for solving problems in coordinate geometry and for various examinations.