Question

What are hyperbolas used for?

Answer 1

Hint: We first explain the mathematical aspect of the conic curve hyperbola. We find the general formula and explain different components of the curve. Then we discuss this conic curve’s use in real-life.

Complete step-by-step solution:
A hyperbola is the mathematical shape in the form of a smooth curve formed by the intersection of two circular cones. The properties of hyperbola allow it to play an important role in the real world where designs and predictions of phenomena are heavily influenced by it.
The general equation hyperbola is \[\dfrac{{{\left( x-\alpha \right)}^{2}}}{{{a}^{2}}}-\dfrac{{{\left( y-\beta \right)}^{2}}}{{{b}^{2}}}=1\].

For the general equation $\left( \alpha ,\beta \right)$ is the centre. The vertices are $\left( \alpha \pm a,\beta \right)$. The coordinates of the foci are $\left( \alpha \pm ae,\beta \right)$. Here \[e=\sqrt{1+\dfrac{{{b}^{2}}}{{{a}^{2}}}}\] is the eccentricity.
Now we talk about this conic curve’s use in real-life.
We see implications of hyperbola in Satellite systems and their path predictions. It is also used in Radio systems’ signals, LORAN, geographic position identification. More commonly known implications are for microscopes, telescopes and televisions.

Note: The hyperbola has an important mathematical equation associated with it - the inverse relation. When an increase in one trait leads to a decrease in another or vice versa which helps in explaining the relationship between the pressure and volume of a gas.