Identify the vertices, foci, and equations for the asymptotes of the hyperbola below.
Type coordinates with parentheses and separated by a comma like this (x,y). If a value is a non-integer, then type as a decimal rounded to the nearest hundredth.
[tex]\frac{(x-1)^2}{9}-\frac{(y-2)^2}{16}=1[/tex]
The vertex with a positive [tex]$x$[/tex] value is the point: (4,2)
The vertex with a negative [tex]$x$[/tex] value is the point: (-2,2)
The foci with a positive [tex]$x$[/tex] value is the point: (6,2)
The foci with a negative [tex]$x$[/tex] value is the point: (-4,2)
One of the asymptotes is the equation [tex]$y=p(x-q)+r$[/tex]
Where the value for [tex]$p$[/tex] is [ ]
Answer (1)
Identify the center ( h , k ) , and the values of a and b from the hyperbola equation.
Determine the equations of the asymptotes using the formula y − k = ± a b ( x − h ) .
Compare the obtained asymptote equation with the given form y = p ( x − q ) + r to find the value of p .
The value of p is 1.33 .
Explanation
Problem Analysis We are given the equation of a hyperbola in the form 9 ( x − 1 ) 2 − 16 ( y − 2 ) 2 = 1 . Our goal is to find the value of p in the equation of one of the asymptotes, which is given as y = p ( x − q ) + r .
Identify Parameters The standard form of a hyperbola centered at ( h , k ) is a 2 ( x − h ) 2 − b 2 ( y − k ) 2 = 1 . Comparing this with the given equation, we have h = 1 , k = 2 , a 2 = 9 , and b 2 = 16 . Therefore, a = 3 and b = 4 .
Asymptote Equations The equations of the asymptotes for a hyperbola in standard form are given by y − k = ± a b ( x − h ) . Substituting the values of h , k , a , and b , we get y − 2 = ± 3 4 ( x − 1 ) .
Isolate y We are given the equation of one of the asymptotes as y = p ( x − q ) + r . Let's consider the asymptote with a positive slope: y − 2 = 3 4 ( x − 1 ) . We can rewrite this as y = 3 4 ( x − 1 ) + 2 .
Determine p Comparing y = 3 4 ( x − 1 ) + 2 with y = p ( x − q ) + r , we can identify the values of p , q , and r . We have p = 3 4 , q = 1 , and r = 2 . The question asks for the value of p .
Final Answer The value of p is 3 4 , which is approximately 1.33 when rounded to the nearest hundredth.
Examples
Understanding hyperbolas and their asymptotes is crucial in various fields like physics and engineering. For instance, the trajectory of a spacecraft during a gravity assist maneuver can be modeled using a hyperbola, with the asymptotes representing the spacecraft's path as it approaches and departs the planet. By knowing the equation of the hyperbola, engineers can precisely calculate the spacecraft's velocity and direction, ensuring a successful mission.
