Question

Auxiliary circle and eccentric angle.

Answer 1

A circle drawn with centre O and transverse axis as diameter
is known as auxiliary circle.

Equation of auxiliary circle is x² + y² = a²

A is any point on the circle whose coordinates are (acosθ,
a sinθ), where Tis known as eccentric angle.

Now, In △OAB, OA = a

\(cos \theta = \frac a{OB} \) ⇒ \(OB = asec\theta\) 

P point lies on hyperbola, so \(\frac{a^2sec^2\theta }{a^2} - \frac{y^2}{b^2} = 1\)

⇒ \(y = \pm b \;tan\theta\)

<!--StartFragment-->P(asecθ, btanθ)

0 ≤  θ < 2π

The equations x = asecθ and y = btanθ represents a hyperbola. So, the parametric form of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2} = 1\) can be represented as x = asecθ, y = btanθ.

For the hyperbola \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} =1\) parametric form is x = h + asecθ ; y = k + btanθ.