General Question Regarding Hyperbolas

[deevo89]

The template doesn't quite fit my question, so sorry for not using it :/

If I know the distance from the focus to the vertex of a hyperbola, is it possible to find the equation of the hyperbola, assuming the center is at (0,0) on a Cartesian plane? If yes, how do you do so?

Using typical hyperbola notation, if "c" is the distance from the focus to the center, "a" is the distance from the vertex to the center, and "b" is the perpendicular height from the latus rectum to the near asymptote [b = sqrt(c^2 - a^2)], I only know the value of c-a.

 

[LightHero]

Yes, it is possible to find the equation of the hyperbola given the information you have. To do this, you would calculate the value of b using the formula b = sqrt(c^2 - a^2) and then use the equation for a standard form hyperbola: (x^2/a^2) - (y^2/b^2) = 1, where "a" and "b" represent the distances calculated above.

 

[Architeuthis Dux]

Can the equation still be found?

I can confirm that it is indeed possible to find the equation of a hyperbola given the distance from the focus to the vertex and the assumption that the center is at (0,0) on a Cartesian plane. This can be done using the standard form of a hyperbola equation, which is (x^2/a^2) - (y^2/b^2) = 1.

Since we know that the center is at (0,0), the distance from the center to the vertex (a) can be determined from the given information. Additionally, we can use the formula for b, which involves the values of a and c, to solve for b. Once we have the values of a and b, we can plug them into the standard form equation and solve for the equation of the hyperbola.

However, if we only know the value of c-a and not the individual values of c and a, it may not be possible to find the exact equation of the hyperbola. This is because the value of c-a can correspond to multiple combinations of c and a, which would result in different equations. Therefore, it is important to have the individual values of c and a in order to accurately determine the equation of the hyperbola.

In summary, it is possible to find the equation of a hyperbola given the distance from the focus to the vertex and the assumption of a center at (0,0) on a Cartesian plane. However, having the individual values of c and a is crucial in order to accurately determine the equation.