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

${\displaystyle \frac{{(x+2)}^2}{4}-\frac{y^2}{4}=1}$

${\displaystyle \frac{{(x+2)}^2}{2^2}-\frac{y^2}{2^2}=1}$

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

So, let x – 2 = X

The obtained equation is of the form

${\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1}$

Where, a = 2 and b = 2

Eccentricity is given by:

${\displaystyle e=\sqrt{1+\frac{b^2}{a^2}}}$

= ${\displaystyle \sqrt{1+\frac{4}{4}}}$

= ${\displaystyle \sqrt{1+1}}$

= ${\displaystyle \sqrt{2}}$

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:

${\displaystyle X=\pm \frac{a}{e}}$

⇒ ${\displaystyle X=\pm \frac{2}{\sqrt{2}}}$

⇒ ${\displaystyle X=\pm \frac{2}{\sqrt{2}}}$

⇒ ${\displaystyle X=\pm \sqrt{2}}$

⇒ ${\displaystyle X\mp \sqrt{2}=0}$

⇒ ${\displaystyle x+2\mp \sqrt{2}=0}$

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