Hyperbola and an ellipse to intersect orthogonally?

In summary, the condition for a hyperbola and an ellipse to intersect orthogonally is that the scalar product of their gradients must be zero. To find the intersection points, the equations of the two curves must be solved simultaneously. Once an intersection point is found, the gradients of each equation can be evaluated at that point. The scalar product of these gradients must be zero. In the given example, the equations of the ellipse and hyperbola are x^2 - y^2 = 1/2 and x^2 + 2y^2 = 2b^2. By solving the equations simultaneously and evaluating the gradients at the intersection point, a constraint for the parameter b can be found.
  • #1
zorro
1,384
0
What is the condition for a hyperbola and an ellipse to intersect orthogonally?
I have a formula for orthogonal circles -> 2g1g2 + 2f1f2 - c1c2 = 0
 
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  • #2


1) Write down the equations of the two curves.

2) Find the intersections by solving the two equations simultaneously.

3) Consider one of theese intersections, say (x0, y0).

4) Derive the first equation with respect to x and y. You get a vector u(x ,y) (the gradient).

5) Do the same with the second equation. Call the gradient v(x, y).

6) Evaluate u and v at the point (x0, y0) calculated earlier.

7) Calculate the scalar product u.v and impose it's zero.
 
  • #3


Thanks for your response.
How do you apply rule 2 if you don't know the equation of one of the curves?
Consider this question-

An ellipse intersects the hyperbola 2x2 - 2y2 = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the co-ordinate axes, then find the equation of the ellipse.

Eccentricity of the ellipse is 1/√2

The relation betwen a and b of ellipse is a2 = 2b2

Now how do you proceed?
 
  • #4


Well if what you say is correct (I didn't check it) then the two equations are:

x^2 - y^2 = 1/2

x^2 + 2y^2 = 2b^2

Try doing steps 2) - 7) with theese, you should get a constraint for the free parameter b.
 
  • #5


The condition for a hyperbola and an ellipse to intersect orthogonally is when the product of their respective slopes (g) and their focal distances (f) is equal to the negative of the product of their respective eccentricities (c). This can be represented by the equation 2g1g2 + 2f1f2 - c1c2 = 0, where the subscripts 1 and 2 represent the hyperbola and ellipse, respectively. This condition ensures that the curves intersect at right angles, forming orthogonal lines. This relationship can also be extended to other conic sections, such as circles and parabolas, where the same equation can be used to determine their orthogonal intersections.
 

FAQ: Hyperbola and an ellipse to intersect orthogonally?

1. What is the difference between a hyperbola and an ellipse?

A hyperbola is a type of conic section that is formed when a plane intersects a double cone, resulting in two separate curved branches. An ellipse, on the other hand, is also a conic section but is formed when a plane intersects a cone at an angle that is less than the angle of the cone's sides. Ellipses have a more rounded shape compared to the more open shape of hyperbolas.

2. How do you know if a hyperbola and an ellipse intersect orthogonally?

A hyperbola and an ellipse intersect orthogonally when their tangent lines at the point of intersection are perpendicular to each other. This means that the slopes of the tangent lines at the point of intersection are negative reciprocals of each other.

3. Can a hyperbola and an ellipse intersect at more than one point?

Yes, it is possible for a hyperbola and an ellipse to intersect at more than one point. This occurs when the two curves have multiple points of tangency with perpendicular tangent lines.

4. What is the significance of orthogonal intersections between a hyperbola and an ellipse?

Orthogonal intersections between a hyperbola and an ellipse have many applications in mathematics and physics. For example, they can be used to determine the orbit of a planet around the sun or to calculate the trajectory of a projectile in physics problems.

5. How do you find the equations of a hyperbola and an ellipse that intersect orthogonally?

To find the equations of a hyperbola and an ellipse that intersect orthogonally, you can use the general equation for each curve and solve for the unknown variables using the condition that the tangent lines at the point of intersection are perpendicular to each other. This will result in a system of equations that can be solved to find the specific equations of the curves.

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