Hyperbola and an ellipse to intersect orthogonally?

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The discussion focuses on the conditions for a hyperbola and an ellipse to intersect orthogonally, emphasizing the need to derive the equations of both curves and find their intersection points. It outlines a method involving the calculation of gradients at the intersection points and checking their scalar product to ensure it equals zero. An example is provided with a specific hyperbola and an ellipse, where the eccentricity of the ellipse is the reciprocal of the hyperbola's eccentricity. The relationship between the semi-major and semi-minor axes of the ellipse is also discussed, leading to the formulation of equations to solve. The conversation encourages applying the outlined steps to derive constraints for the parameters involved.
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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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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.
 


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?
 


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.
 
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