Question

Find the centre, eccentricity, foci and directrice of the hyperbola.

 x² − y² + 4x = 0

Answer 1

Given:

The equation ⇒ x² – y² + 4x = 0

Let us find the centre, eccentricity, foci and directions of the hyperbola

By using the given equation

x² – y² + 4x = 0

x² + 4x + 4 – y² – 4 = 0

(x + 2)² – y² = 4

`(x + 2)^2/4 - y^2/4 = 1`

`(x + 2)^2/2^2 - y^2/2^2 = 1`

Here, center of the hyperbola is (2, 0)

So, let x – 2 = X

The obtained equation is of the form

`x^2/a^2 - y^2/b^2 = 1`

Where, a = 2 and b = 2

Eccentricity is given by:

`e = sqrt(1 + b^2/a^2)`

= `sqrt(1 + 4/4)`

= `sqrt(1 + 1)`

= `sqrt2`

Foci: The coordinates of the foci are (± ae, 0)

X = ± 2√2 and Y = 0

X + 2 = ± 2√2 and Y = 0

X= ± 2√2 – 2 and Y = 0

So, Foci = (± 2√2 – 2, 0)

Equation of directrix are:

`X = pm a/e`

⇒ `X = pm 2/sqrt2`

⇒ `X = pm 2/sqrt2`

⇒ `X = pm sqrt2`

⇒ `X mp sqrt2 = 0`

⇒ `x + 2 mp sqrt2 = 0`

x + 2 − √2 = 0 and x + 2 + √2 = 0

∴ The center is (−2, 0), eccentricity (e) = √2, Foci = (−2 ± 2√2, 0), Equation of directrix = x + 2 = ±√2