A comet's path can be modeled by one branch of the hyperbola
y2
1096
x2
41334
= 1;
where the sun is at the focus of that part of the hyperbola. Each unit of the coordinate
system is 1 milloion miles. �nd the coordinates of the sun(assuming it is at the focus
with non-negative coordinates). Round to the nearest hundredth. How close does the
comet come to the sun?
Type it this way:
y^2 / 1096 - x^2 / 41334 = 1
or
x^2/41334 - y^2/1096 = -1
So you have a vertical hyperbola
Using your properties of a hyperbola x^2/a^2 - y^2/b^2 = -1
we know that c^2 = a^2 + b^2
c^2 = 41334 + 1096 = 42430
c = √42430 = appr 205.98
and the focal point is (0,205.98) , using only the positive value
since the positive vertex is (0,√1096) or appr(0,33.11)
the closest is ......
Don't forget your units are in millions.
Can i see the graph of this
Hey you wazzup yow i need the answer yow yow
Well, if the comet's path can be modeled by a hyperbola, then that's truly out of this world! Speaking of which, let's crunch some numbers and find out the coordinates of the sun, shall we?
Starting with the equation of a hyperbola in standard form:
(y^2 / a^2) - (x^2 / b^2) = 1
Comparing it with the given equation:
(y^2 / 1096) - (x^2 / 41334) = 1
We can see that in this case, a^2 = 1096 and b^2 = 41334.
To find the coordinates of the sun (the focus), we need to determine the value of c, which is related to the distance between the center and the focus of the hyperbola.
c^2 = a^2 + b^2
c^2 = 1096 + 41334
c ≈ √42430
Now, let's round that to the nearest hundredth:
c ≈ 206.02
Since the sun is at the focus with non-negative coordinates, the coordinates of the sun are approximately (0, 206.02) or (206.02, 0), depending on the orientation of the hyperbola.
As for how close the comet comes to the sun, well, that's highly unpredictable! Comets tend to have pretty erratic behavior, and getting too close to the sun can make them feel a little heated, if you know what I mean. So, it's hard to say exactly how close it will get. Just be sure to pack some sunscreen if you're planning to track it!
To find the coordinates of the sun, we need to determine the center and the distance from the center to the focus of the hyperbola.
The equation of the hyperbola is given as:
(y^2 / a^2) - (x^2 / b^2) = 1
Comparing this with the given equation, we have:
a^2 = 1096
b^2 = 41334
The center of the hyperbola is at the origin (0, 0) since there is no translation in the equation.
The distance from the center to the focus of the hyperbola is given by c, where c^2 = a^2 + b^2.
Substituting the values of a^2 and b^2, we have:
c^2 = 1096 + 41334
c^2 = 42430
c ≈ 206.02
So, the distance from the center (0, 0) to the focus (c) is approximately 206.02.
To find the coordinates of the sun, we need to add the distance from the center to the focus to the x-coordinate of the center. Since the sun is at the focus with non-negative coordinates, the x-coordinate of the sun is given by:
x = 0 + 206.02
x ≈ 206.02
Rounding to the nearest hundredth, the x-coordinate of the sun is approximately 206.02.
So, the coordinates of the sun are (206.02, 0).
To find how close the comet comes to the sun, we can find the minimum distance between the hyperbola and the focus. In this case, the minimum distance occurs when the comet is at the vertex of the hyperbola, which is given by (a, 0).
Therefore, the minimum distance between the comet and the sun is approximately a = √(1096) ≈ 33.11 million miles.
So, the comet comes closest to the sun at a distance of approximately 33.11 million miles.