Question

An asteroid follows a hyperbolic path defined by the equation 16x^2 − 9y^2 = 576.

If one of its foci is the sun’s position, the minimum distance between the asteroid and the sun is(how many?) AU. (Note that x and y are expressed in astronomical units, AU.)

Answer 1

not a pro with these
related to a 3-4-5 triangle
do you happen to have choices?

Answer 2

No I don't :(

Answer 3

@iambatman can you help please?

Answer 4

@ganeshie8 ?

Answer 5

@TuringTest @wio

Answer 6

@jim_thompson5910 @Directrix @PRAETORIAN.10

Answer 7

As a start : change the given equation into standard form
\[\large \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2}=1\]

Answer 8

divide both sides by 576 so that the right hand side becomes 1

Answer 9

ok

Answer 10

@Jhannybean

Answer 11

\[16x^2 − 9y^2 = 576\]
dividing \(576\) through out you get
\[\dfrac{16x^2}{576} − \dfrac{9y^2}{576} = \dfrac{576}{576}\]

Answer 12

which is same as
\[\dfrac{x^2}{36} - \dfrac{y^2}{64} = 1\]

Answer 13

compare this with the standard form and see if you can eyeball the values of \(a\) and \(b\)

Answer 14

6 & 8?

Answer 15

Yes!
a = 6
b = 8

so the x coordinate of vertex is 6

Answer 16

Next find the x coordinate of focus by using the relation :
\[\large c^2 = a^2+b^2\]

Answer 17

100

Answer 18

\[\large c^2 = a^2+b^2\]
\[\large c^2 = 6^2+8^2\]
\[\large c^2 = 100\]
\[\large c = ?\]

Answer 19

10

Answer 20

correct!