# Definition of Ellipse

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.

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The Ellipse