# Length of Latus Rectum of Ellipse Formula

Here you will learn what is the formula for the length of latus rectum of ellipse with examples..

Let’s begin –

## Length of Latus Rectum of Ellipse

#### (i) For the ellipse $$x^2\over a^2$$ + $$y^2\over b^2$$ = 1, a > b

Length of the Latus Rectum = $$2b^2\over a$$

Equation of latus rectum is x = $$\pm ae$$.

#### (ii) For the ellipse $$x^2\over a^2$$ + $$y^2\over b^2$$ = 1, a < b

Length of the Latus Rectum = $$2a^2\over b$$

Equation of latus rectum is y = $$\pm be$$.

Example : For the given ellipses, find the length of latus rectum.

(i)  $$16x^2 + 25y^2$$ = 400

(ii)  $$x^2 + 4y^2 – 2x$$ = 0

Solution :

(i)  We have,

$$16x^2 + 25y^2$$ = 400 $$\implies$$ $$x^2\over 25$$ + $$y^2\over 16$$,

where $$a^2$$ = 25 and $$b^2$$ = 16 i.e. a = 5 and b = 4

Clearly a > b,

Therefore, Length of the Latus Rectum (L) = $$2b^2\over a$$

$$\implies$$  L = $$2\times 16\over 5$$ = $$32\over 5$$

(ii) We have,

$$x^2 + 4y^2 – 2x$$ = 0

$$\implies$$ $$(x – 1)^2$$ + 4$$(y – 0)^2$$ = 1

$$\implies$$  $$(x – 1)^2\over 1^2$$ + $$(y – 0)^2\over (1/2)^2$$ = 1

Here, a = 1 and b = 1/2

Clearly a > b,

Therefore, Length of the Latus Rectum (L) = $$2b^2\over a$$

$$\implies$$  L = $$2\times 1/4\over 1$$ = $$1\over 2$$

Note : For the ellipse $$(x – h)^2\over a^2$$ + $$(y – k)^2\over b^2$$ = 1 with center (h. k),

(i) For ellipse a > b,

The equation of latus rectum is x = $$\pm ae + h$$.

(ii) For ellipse a < b,

The equation of latus rectum is y = $$\pm be + k$$.