We will discuss the definition of ellipse and how to find the equation of the ellipse whose focus, directrix and eccentricity are given.

An ellipse is the locus of a point P moves on this plane in such a way that its distance from the fixed point S always bears a constant ratio to its perpendicular distance from the fixed line L and if this ratio is less than unity.

An ellipse is the locus of a point in a plane which moves in the plane in such a way that the ratio of its distance from a fixed point (called focus) in the same plane to its distance from a fixed straight line (called directrix) is always constant which is always less than unity.

The constant ratio usually denoted by e (0 < e < 1) and is known as the eccentricity of the ellipse.

If S is the focus, ZZ' is the directrix and P is any point on the
ellipse, then by definition

\(\frac{SP}{PM}\) = e

⇒ SP = e ∙ PM

The fixed point S is called a Focus and the fixed straight line L the corresponding Directrix and the constant ratio is called the Eccentricity of the ellipse.

Solved example to find the equation of the ellipse whose focus, directrix and eccentricity are given:

Determine the equation of the ellipse whose focus is at (-1, 0), directrix is 4x + 3y + 1 = 0 and eccentricity is equal to \(\frac{1}{√5}\).

**Solution:**

Let S (-1, 0) be the focus and ZZ' be the directrix. Let P (x, y) be any point on the ellipse and PM be perpendicular from P on the directrix. Then by definition

SP = e.PM where e = \(\frac{1}{√5}\).

⇒ SP\(^{2}\) = e\(^{2}\) PM\(^{2}\)

⇒ (x + 1)\(^{2}\) + (y - 0)\(^{2}\) = \((\frac{1}{\sqrt{5}})^{2}[\frac{4x + 3y + 1}{\sqrt{4^{2} + 3^{2}}}]\)

⇒ (x + 1)\(^{2}\) + y\(^{2}\) = \(\frac{1}{25}\)\(\frac{4x + 3y + 1}{5}\)

⇒ x\(^{2}\) + 2x + 1 + y\(^{2}\) = \(\frac{4x + 3y + 1}{125}\)

⇒ 125x\(^{2}\) + 125y\(^{2}\) + 250x + 125 = 0, which is the required equation of the ellipse.

**● ****The Ellipse**

**Definition of Ellipse****Standard Equation of an Ellipse****Two Foci and Two Directrices of the Ellipse****Vertex of the Ellipse****Centre of the Ellipse****Major and Minor Axes of the Ellipse****Latus Rectum of the Ellipse****Position of a Point with respect to the Ellipse****Ellipse Formulae****Focal Distance of a Point on the Ellipse****Problems on Ellipse**

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