Question

What is $\text{'}a\text{'}$ in a hyperbola?

Answer 1

Hyperbola:

Hyperbola is formed by a set of points moving so that the absolute difference in the distance between two fixed points remains constant.

It is formed when a vertical plane intersects a double cone.

The general equation of hyperbola opening left and right is $\frac{{\left(x-h\right)}^2}{a^2}-\frac{{\left(y-k\right)}^2}{b^2}=1$, where $\left(h,k\right)$ are the coordinates of the center and the vertices are $\left(h\pm a,k\right)$

The general equation of hyperbola opening upwards and downwards is $\frac{{\left(y-k\right)}^2}{b^2}-\frac{{\left(x-h\right)}^2}{a^2}=1$, where $\left(h,k\right)$ are the coordinates of the center and the vertices are $\left(h,k\pm b\right)$

The shape of the hyperbola is determined by the asymptotes. The asymptotes pass through the center and its slope is $\pm \frac{b}{a}$.

For any given hyperbola, the transverse axis is the imaginary line joining the two vertices and the conjugate axis is the line bisecting the distance between the two vertices and perpendicular to the transverse axis.

The rectangle formed by the transverse and conjugate axes is called the fundamental rectangle, of which the diagonals are the asymptotes.

Hence, the value of $a$ is the distance from any of the vertex of the hyperbola to the center.