1. ## Hyperbola exercise

can someone help me with this exercise ? So far , I only know that the first element of the equation is 5.

If the asymptotes of the hyperbolic path of a particle are given by y = + - 3/4x and the closest the particle comes to the nucleus is 5 units , find the equation of the particle's path.

* see attachment

2. Hello, Vance!

If the asymptotes of the hyperbolic path of a particle are given by $\displaystyle y = \pm\tfrac{3}{4}x$
and the closest the particle comes to the nucleus (origin) is 5 units,
find the equation of the particle's path.

From the asymptotes, we have: .$\displaystyle \frac{b}{a} \:=\:\frac{3}{4} \quad\Rightarrow\quad b \:=\:\tfrac{3}{4}a$

Since $\displaystyle a = 5$, we have: .$\displaystyle b \:=\:\tfrac{3}{4}(5) \:=\:\tfrac{15}{4}$

The equation of the entire hyperbola is: .$\displaystyle \frac{x^2}{5^2} - \frac{y^2}{(\frac{15}{4})^2} \;=\;1$

We want only the right branch: .$\displaystyle \frac{x^2}{25} - \frac{16y^2}{225} \;=\;1\;\;\text{ for }x \ge 5$