## Ellipse in standard form concept

Let **F(-c, 0)** be the focus and the line** x = -c / e ²** be the directrix of an ellipse with the **eccentricity e** that is** (0, e < 1).** Let** p(x, y)** be any point on the ellipse and suppose that **|PM|** is the** perpendicular** **distance of P** from the directrix then.

**|PM| = x + c / e ²**

The condition **|PF| = e |PM|**

**Analytic form:**

**(x+c)² +y² = e² (x + c / e ²)²**

**x²+2.c.x+c² + y²= e²x²+2 c.x + c ² / e²**

**x ² (1 – e ²) + y² = c² / e ²(1 – e²)**

**x ²(1 – e²) + y ² = a ² (1 – e ²) **

** where c / e = a**

**x ² / a ²+ y² / a² (1 – e²) = 1………………… A**

**if we write**

** b² = a ² (1 – e²) **

**then equation A become**

………………… B

**Which is an equation of standard form.**

### Eccentricity of the ellipse is e = c / a

We have

** b² = a ² (1 – e²)**

** = a ² – a ² e² **

** = a ² – c ²**

**Eccentricity when b < a, of ellipse:**

**(i)** from, the relation **b² = a ² (1 – e²)** we note that **b < a**

**(ii)** since we have **c / e = a ,** the focus F has coordinated **(- a.e, 0)** and the equation

**of the DirectX is x = – a / e**

**Foci of an ellipse:**

**(iii)** If we take the point **(a.e, 0) **as focus and the line **x = a / e** as DirectX, it can be seen easily

that, we again obtained equation A. thus the standard form of ellipse has two **Foci (- a.e, 0) and (a.e, 0)**

and two directrix **X = ± a / e**

#### Parametric form of an ellipse from equation B:

**ellipse in standard form more concept:**

The point **(a cos θ, b sin θ)** lies on equation A for all real θ

Where

**X = a cos θ**

**Y = b sin θ**

If we put **b = a, ** in equation B then it becomes

**x² + y² = a ²**

Which is a circle. In this case, **b² = a ² (1 – e²) = a ² and e = 0.**

Thus, a** circle is a special case** of an ellipse with **eccentricity 0** and Foci tending to the circle