Find the equation of hyperbola

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In summary, from the given information, a pair of tangents are drawn from the point (2√2,1) to a hyperbola with equation \dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} = 1. These tangents intersect the coordinate axes in concyclic points. One of the tangents is inclined at an angle of tan^{-1}\frac{1}{√2} with the transverse axis of the hyperbola. The equation of the hyperbola and the circle formed by the concyclic points cannot be determined without additional information.
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utkarshakash
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Homework Statement


From the point (2√2,1) a pair of tangents are drawn to [itex]\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2} = 1,[/itex] which intersect the coordinate axes in concyclic points . If one of the tangents is inclined at an angle of [itex]tan^{-1}\frac{1}{√2}[/itex] with the transverse axis of the hyperbola , then find the equation of
i) hyperbola ii)circle formed using concyclic points

Homework Equations



The Attempt at a Solution


Equation of tangent
[itex]y=mx+\sqrt{a^2m^2-b^2}[/itex]
where m = [itex]tan^{-1}\frac{1}{√2}[/itex]
Passing it through the given point will give me an equation in a and b. But there are two unknowns.
 
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  • #2
Condition for tangency is ##c^2=a^2m^2-b^2##. Can you find c?

edit: sorry didn't see that you already did this. I think you have to use the fact that the points are concyclic, or find the equation of the other tangent.
 
Last edited:
  • #3
MrWarlock616 said:
Condition for tangency is ##c^2=a^2m^2-b^2##. Can you find c?

edit: sorry didn't see that you already did this. I think you have to use the fact that the points are concyclic, or find the equation of the other tangent.

But how?
 
  • #4
utkarshakash said:
But how?

I think some data might be missing. All I can think of is that the slope of the tangent is -B/A where A and B represent the intercepts of this tangent. A relation between A and B can be found by looking at the circle formed by the points of the intercept, as they are at equal distances from the centre...but the centre is not known too..
 

Related to Find the equation of hyperbola

1. What is a hyperbola?

A hyperbola is a type of mathematical curve that is formed when a plane cuts through a double cone at an angle. It is defined by two foci and a constant difference between the distances from any point on the curve to the two foci.

2. How do I find the equation of a hyperbola?

The standard equation for a hyperbola is (x-h)^2/a^2 - (y-k)^2/b^2 = 1, where (h,k) represents the coordinates of the center of the hyperbola, and a and b are the distances from the center to the vertices. To find the equation of a hyperbola, you will need to know the coordinates of the center, the distances from the center to the vertices, and the orientation of the hyperbola.

3. What is the difference between a horizontal and a vertical hyperbola?

A horizontal hyperbola has its transverse axis (the line passing through the center and the two vertices) parallel to the x-axis, while a vertical hyperbola has its transverse axis parallel to the y-axis. This affects the placement of the a and b values in the standard equation, as well as the direction of the curve.

4. How many solutions can a hyperbola have?

A hyperbola can have an infinite number of solutions, as it is a continuous curve. However, when solving for specific points or values, a hyperbola can have up to four solutions (two on each branch of the curve).

5. What are the applications of hyperbolas in real life?

Hyperbolas have many practical applications in fields such as astronomy, engineering, and physics. They can be used to model the orbits of planets, the paths of satellites, and the trajectories of projectiles. They are also used in the design of parabolic reflectors, such as satellite dishes and telescopes.

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