Question

For the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$, the value of $a$is __ , the value of b is ___ , and the transverse axis is the ___ -axis.

Short answer

For the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$, the value of $a$is $2$, the value of $b$is $3$, and the transverse axis is the $x-$axis.

Step-by-step solution

Step 1. Given Information

The given equation is$\frac{x^2}{4}-\frac{y^2}{9}=1$.

Step 2. Objective of the Problem

We need to find the value of $a,$and $b$and also find the transverse axis.

Step 3. Concept 

An equation of the hyperbola with centre at $(0,0)$, foci at $(-c,0)$and $(c,0)$,

and vertices at $(-a,0)$and $(a,0)$is

$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $b^2=c^2-a^2$

Here the transverse axis is the $x-$axis.

Step 4. Finding the value of a,b and the transverse axis.

The given equation $\frac{x^2}{4}-\frac{y^2}{9}=1$is of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $a^2=9$and $b^2=4$.

We have $a^2=4.$This implies, $a=\sqrt{4}=2$.

Further, we have $b^2=9.$This implies, $b=\sqrt{9}=3$.

Further, the transverse axis is the$x-$axis.