Solving for Focus Points on a Hyperbole Formula | x^2-3y^2=3

[transgalactic]

the formula of the hyperbole is x^2 -3*y^2=3.

find a dot on the hyperbole that the sum of the distances
between the dot and the focus points equals to 6

i devided all the formula by 3 in order to find a^2 ,b^2
i found the focus point by the formula c^2=a^2 +b^2

the problem is the the definision of the hyperbole
is that the subtraction of distances from every point to
the focus point equals to 2a

there is no law about the sum of the distances?

only the law i mentioned before deals with them

the answer in the book (+-3:+-2)

please help

 

[Dick]

Sure. Sum of the distances=6 would define an ellipse. So you want to find a point that is both on this ellipse and your original hyperbola. Sounds like an intersection problem to me.

 

[transgalactic]

no ellipse involved i know that a sum of distances is an ellipse definition

but there is no ellipse here

 

[Dick]

My suggestion was to put an ellipse into the problem! The one defined by sum of the focal distances = 6. That will give you a second equation in x,y to intersect with your hyperbola.

 

[transgalactic]

i didnt understand
if i will create such an ellipse ,
it will mean that 2a=6 a=3
thats will leave me B unknown for this ellipse??

and even if i some how find B
does these shared points will be these points we are looking for??

 

[Dick]

You know the foci of the ellipse (they are the same as the hyperbola). You know the sum of the focal distances (6). Isn't that enough to write down an equation for the ellipse?

 

[transgalactic]

even if we put the same foci and we will find this ellipse
why the should shared points of the given hyperbole and this ellipse
represent the points we are looking for??

 

[Dick]

The shared points will be both on the hyperbola and have the sum of the focal distances = 6 (since they are on the ellipse). Isn't that what you want?

 

[transgalactic]

ahhhhhhhh ok
thanks

 

[Curious3141]

This is the zillionth time I've seen hyperbola misspelled hyperbole! The zillionth I tells ya!

 

[transgalactic]

in the dictionary it says hyperbole

 

[Dick]

There are two different words. Hyperbola is the geometric figure, hyperbole (pronounced like hyper-bolee) means an exaggeration or overstatement. French/latin etc use hyperbole for the geometric figure. Hyperbolic can refer to both. Fascinating!

 

[Dick]

BTW, a statement can be elliptical or circular as well. Don't be confused!