Question

A point on the curve \[\frac{x^2}{\mathrm{A}^2}-\frac{y^2}{\mathrm{B}^2}=1\] is______.

Options

• \[(A\cos\theta,B\sin\theta)\]

• \[(Asec\theta,Btan\theta)\]

• \[(A\cos^2\theta,B\sin^2\theta)\]

• None of these

Answer 1

A point on the curve \[\frac{x^2}{\mathrm{A}^2}-\frac{y^2}{\mathrm{B}^2}=1\] is \[(Asec\theta,Btan\theta)\].

Explanation:

The hyperbola is \[\frac{x^{2}}{A^{2}}-\frac{y^{2}}{B^{2}}=1.\]

A point lies on it only if substituting satisfies the equation. Using \[x=A\sec\theta,y=B\tan\theta{:}\]

This works because of the standard trigonometric identity \[\sec^{2}\theta-\tan^{2}\theta=1\], which mirrors the structure of the hyperbola equation exactly.