Hyperbola and an ellipse to intersect orthogonally?

[zorro]

What is the condition for a hyperbola and an ellipse to intersect orthogonally?
I have a formula for orthogonal circles -> 2g₁g₂ + 2f₁f₂ - c₁c₂ = 0

 

[Petr Mugver]

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.

 

[zorro]

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?

 

[Petr Mugver]

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.

 

[CellCoree]

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.