The value of $a$ , $b$ , $c$ , $e$ and the coordinates of the center of the Sun if the center of the hyperbolic orbit is $0,0$ and the Sun lies on the positive $x$ axis.

It has been determined that $a=1440$ , $b=600$ , $c=1560$ and $e≈1.083$ .

It has also been determined that the coordinates of the center of the Sun are $1560,0$ .

Given:

A comet following a hyperbolic path about the Sun has a perihelion of $120$ Gm. When the line from the comet to the Sun is perpendicular to the focal axis of the orbit, the comet is $250$ Gm from the Sun.

Concept used:

The perihelion distance of a hyperbola is $c−a$ .

Calculation:

It is given that a comet follows a hyperbolic path about the Sun such that the center of the hyperbolic orbit is $0,0$ and the Sun lies on the positive $x$ axis.

Then, the equation of the hyperbola is given as $x2a2−y2b2=1$ .

As discussed, the perihelion distance of a hyperbola is $c−a$ .

It is given that the comet following a hyperbolic path about the Sun has a perihelion of $120$ Gm.

Then, $c−a=120$ .

Now, for a hyperbola, $c2=a2+b2$ .

Put $x=c$ in $x2a2−y2b2=1$ to get,

$c2a2−y2b2=1y2b2=c2a2−1y2=b2c2a2−1$

Put $c2=a2+b2$ in $y2=b2c2a2−1$ to get,

$y2=b2a2+b2a2−1=b21+b2a2−1=b2b2a2=b4a2$

So, $y=±b2a$ .

It is given that when the line from the comet to the Sun is perpendicular to the focal axis of the orbit, the comet is $250$ Gm from the Sun.

This implies that $b2a=250$ .

Simplifying,

$b2=250a$

Put $b2=250a$ in $c2=a2+b2$ to get,

$c2=a2+250a$

Simplifying,

$c2−a2=250a$

On further simplification,

$c−ac+a=250a$

Put $c−a=120$ in the above equation to get,

$120c+a=250a$

Simplifying,

$120c+120a=250a$

On further simplification,

$120c=130a$

Now, $c−a=120$ . Then, $c=120+a$ .

Put $c=120+a$ in $120c=130a$ to get,

$120120+a=130a14400+120a=130a10a=14400a=1440$

So, $a=1440$ .

Put $a=1440$ in $b2=250a$ to get,

$b2=2501440$

Simplifying,

$b=±600$

Taking the positive value, $b=600$ .

Put $a=1440$ in $c=120+a$ to get,

$c=1560$

Put $c=1560$ and $a=1440$ in $e=ca$ to get,

$e=15601440$

Simplifying,

$e≈1.083$

Now, the Sun is at the focus, which is the point $c,0$ .

Then, the coordinates of the center of the Sun are $1560,0$ .

Conclusion:

It has been determined that $a=1440$ , $b=600$ , $c=1560$ and $e≈1.083$ .

It has also been determined that the coordinates of the center of the Sun are $1560,0$ .

Chapter 8.3, Problem 54E

Chapter 8.3, Problem 56E

Chapter 8 Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

Ch. 8.1 - Prob. 1QR Ch. 8.1 - Prob. 2QR Ch. 8.1 - Prob. 3QR Ch. 8.1 - Prob. 4QR Ch. 8.1 - Prob. 5QR Ch. 8.1 - Prob. 6QR Ch. 8.1 - Prob. 7QR Ch. 8.1 - Prob. 8QR Ch. 8.1 - Prob. 9QR Ch. 8.1 - Prob. 10QR

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - Prob. 37E Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - Prob. 43E Ch. 8.1 - Prob. 44E Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.1 - Prob. 47E Ch. 8.1 - Prob. 48E Ch. 8.1 - Prob. 49E Ch. 8.1 - Prob. 50E Ch. 8.1 - 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Prob. 6QR Ch. 8.4 - Prob. 7QR Ch. 8.4 - Prob. 8QR Ch. 8.4 - Prob. 9QR Ch. 8.4 - Prob. 10QR Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - Prob. 6E Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - Prob. 17E Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 8.4 - Prob. 27E Ch. 8.4 - Prob. 28E Ch. 8.4 - Prob. 29E Ch. 8.4 - Prob. 30E Ch. 8.4 - Prob. 31E Ch. 8.4 - Prob. 32E Ch. 8.4 - Prob. 33E Ch. 8.4 - Prob. 34E Ch. 8.4 - Prob. 35E Ch. 8.4 - Prob. 36E Ch. 8.4 - Prob. 37E Ch. 8.4 - Prob. 38E Ch. 8.4 - Prob. 39E Ch. 8.4 - Prob. 40E Ch. 8.4 - Prob. 41E Ch. 8.4 - Prob. 42E Ch. 8.4 - Prob. 43E Ch. 8.4 - Prob. 44E Ch. 8.4 - Prob. 45E Ch. 8.4 - Prob. 46E Ch. 8.4 - 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Prob. 1E Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - Prob. 6E Ch. 8.6 - Prob. 7E Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - Prob. 11E Ch. 8.6 - Prob. 12E Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - Prob. 15E Ch. 8.6 - Prob. 16E Ch. 8.6 - Prob. 17E Ch. 8.6 - Prob. 18E Ch. 8.6 - Prob. 19E Ch. 8.6 - Prob. 20E Ch. 8.6 - Prob. 21E Ch. 8.6 - Prob. 22E Ch. 8.6 - Prob. 23E Ch. 8.6 - Prob. 24E Ch. 8.6 - Prob. 25E Ch. 8.6 - Prob. 26E Ch. 8.6 - Prob. 27E Ch. 8.6 - Prob. 28E Ch. 8.6 - Prob. 29E Ch. 8.6 - Prob. 30E Ch. 8.6 - Prob. 31E Ch. 8.6 - Prob. 32E Ch. 8.6 - Prob. 33E Ch. 8.6 - Prob. 34E Ch. 8.6 - Prob. 35E Ch. 8.6 - Prob. 36E Ch. 8.6 - Prob. 37E Ch. 8.6 - Prob. 38E Ch. 8.6 - Prob. 39E Ch. 8.6 - Prob. 40E Ch. 8.6 - Prob. 41E Ch. 8.6 - Prob. 42E Ch. 8.6 - Prob. 43E Ch. 8.6 - Prob. 44E Ch. 8.6 - Prob. 45E Ch. 8.6 - Prob. 46E Ch. 8.6 - Prob. 47E Ch. 8.6 - Prob. 48E Ch. 8.6 - Prob. 49E Ch. 8.6 - Prob. 50E Ch. 8.6 - Prob. 51E Ch. 8.6 - 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