Question

experiments are performed in which moving particles are deflected by various forces. the particles are eventually deflected along hyperbolic paths. looking at the given figure, if a particle gets as close as 7 units to the nucleus along a hyperbolic path with an asymptote given by $y = \frac{1}{9}x$, what is the equation of its path? the equation of the hyperbolic path is \boxed{}. (type your answer in standard form. use integers or fractions for any numbers in the equation.)

Explanation:

Step1: Identify hyperbola type

Right-opening hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Asymptote: $y = \frac{b}{a}x = \frac{1}{9}x \Rightarrow \frac{b}{a} = \frac{1}{9} \Rightarrow b = \frac{a}{9}$

Step2: Closest distance to origin

Closest point is $(a,0)$; distance = $a = 7 \Rightarrow a = 7$

Step3: Calculate $b^2$

$b = \frac{7}{9} \Rightarrow b^2 = \frac{49}{81}$

Step4: Write standard equation

Substitute $a^2=49$, $b^2=\frac{49}{81}$:
$\frac{x^2}{49} - \frac{y^2}{\frac{49}{81}} = 1$
Simplify: $\frac{x^2}{49} - \frac{81y^2}{49} = 1$
Multiply by 49: $x^2 - 81y^2 = 49$

Answer:

$x^2 - 81y^2 = 49$