Parametric Form and Auxiliary Circle

Parametric Form and Auxiliary Circle: Ellipse Essentials for JEE Main

Hello JEE aspirants! In this lesson, we'll explore the parametric form of the ellipse, a powerful tool that simplifies many problems related to ellipses in coordinate geometry. We'll also delve into the auxiliary and director circles, uncovering their geometric significance and how they relate to the ellipse. Understanding these concepts will give you an edge when tackling JEE Main questions.

1. Parametric Representation: A New Perspective on the Ellipse

So far, we've primarily described the ellipse using its Cartesian equation: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$. But there's another way to represent points on the ellipse, using a parameter. This is particularly useful when dealing with tangents, normals, and other geometric properties.

The Parametric Form: Any point $P(x, y)$ on the ellipse can be represented as:

where $\theta$ is a parameter known as the eccentric angle.

Geometric Interpretation: Imagine a circle with radius $a$ (the semi-major axis) centered at the origin. This is the auxiliary circle, which we will discuss in detail later. For any point $P$ on the ellipse, extend a vertical line until it intersects the auxiliary circle at point $Q$. The angle that $OQ$ makes with the positive x-axis is the eccentric angle $\theta$ of point $P$. In other words, drop a perpendicular from any point on the ellipse to the major axis, extend it to the auxiliary circle. Now, the angle subtended by this point on the auxiliary circle with the positive x-axis is the eccentric angle.

Let's say $a=5$ and $b=3$. If $\theta = \frac{\pi}{3}$, then the corresponding point on the ellipse is $x = 5\cos(\frac{\pi}{3}) = \frac{5}{2}$ and $y = 3\sin(\frac{\pi}{3}) = \frac{3\sqrt{3}}{2}$. So, the point is $\left(\frac{5}{2}, \frac{3\sqrt{3}}{2}\right)$.

2. The Eccentric Angle: Decoding the Parameter

The eccentric angle, denoted by $\theta$, is crucial for understanding the parametric form. It's not the angle that the line joining the origin to the point on the ellipse makes with the x-axis (except when $a = b$, i.e., the ellipse becomes a circle!). As described above, it is the angle made by the corresponding point on the auxiliary circle. A very important coordinate is $(a\cos\theta, b\sin\theta)$, note that this ellipse can be generated from the unit circle via two scalings.

3. Auxiliary Circle: The Guiding Circle

The auxiliary circle is a circle drawn with the major axis of the ellipse as its diameter. It's centered at the center of the ellipse and has a radius equal to the semi-major axis, $a$.

Equation of the Auxiliary Circle:

The auxiliary circle is instrumental in defining the eccentric angle, as we've seen. It's also useful in visualizing and solving problems related to tangents and normals to the ellipse.

4. Director Circle: Locus of Perpendicular Tangents

The director circle is the locus of the point of intersection of two perpendicular tangents to the ellipse.

Equation of the Director Circle:

Geometric Meaning: Imagine drawing tangents to the ellipse. Now imagine two tangent lines that meet at a right angle. If you take the set of all possible intersections of such tangents, you trace a circle around the ellipse, and that circle is the Director Circle.

5. Focal Chord Properties: Relationships with the Foci

A focal chord is a chord of the ellipse that passes through either of the foci. Let $P$ and $Q$ be the endpoints of a focal chord.

Property 1:

where $S$ is a focus, and $P$ and $Q$ are the ends of the focal chord. $a$ and $b$ are the semi-major and semi-minor axes, respectively.

Property 2: Sum of Focal Distances For any point $P$ on the ellipse:

where $S$ and $S'$ are the two foci of the ellipse. This is a defining property of the ellipse.

Mastering these concepts – parametric representation, eccentric angle, auxiliary and director circles, and focal chord properties – will significantly improve your problem-solving skills for the JEE Main. Keep practicing, and good luck!