Question

If the vertices and foci of a hyperbola are respectively \((\pm 3,0)\) and \((\pm 4,0)\), then the parametric equations of that hyperbola are

• A. \(x=3\sec\theta,\;y=7\tan\theta\)
• B. \(x=\sqrt{3}\sec\theta,\;y=\sqrt{7}\tan\theta\)
• C. \(x=\sqrt{3}\sec\theta,\;y=7\tan\theta\)
• D. \(x=3\sec\theta,\;y=\sqrt{7}\tan\theta\)

Hint:

For the hyperbola \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1\), vertices are \((\pm a,0)\), foci are \((\pm c,0)\), and \(c^2=a^2+b^2\).

Answer 1

The Correct Option is D

Step 1: Identify the orientation.
The vertices $(\pm3,0)$ lie on the $x$-axis, so the transverse axis is horizontal and the hyperbola is $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.
Step 2: Find $a$.
The vertices are $(\pm a,0)$, so $a = 3$ and $a^2 = 9$.
Step 3: Find $c$.
The foci are $(\pm c,0) = (\pm4,0)$, so $c = 4$ and $c^2 = 16$.
Step 4: Find $b$.
For a hyperbola, $b^2 = c^2 - a^2 = 16 - 9 = 7$, so $b = \sqrt7$.
Step 5: Write the parametric equations.
The standard parametrisation of $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is $x = a\sec\theta$, $y = b\tan\theta$.
Step 6: Substitute the values.
With $a=3$ and $b=\sqrt7$, the equations are $x = 3\sec\theta$, $y = \sqrt7\tan\theta$.
\[ \boxed{x=3\sec\theta,\; y=\sqrt{7}\tan\theta} \]