haruspex said:Try writing the generic equation for a normal to an ellipse with same centre and axes.
I assume you meant [itex]a'sec \phi x - b'cosec \phi y = a'^2 - b'^2[/itex]utkarshakash said:[itex]a'sec \phi x - b'cosec \phi y = a^2 - b^2[/itex]
haruspex said:I assume you meant [itex]a'sec \phi x - b'cosec \phi y = a'^2 - b'^2[/itex]
It remains to find expressions for a', b' and phi in terms of a, b and θ that make this the same as the equation for MN. There is a constraint regarding which of a', b' and phi can depend on which of a, b and θ.
Yes. You want to make a′x sec(ϕ)−b′y cosec(ϕ)=a′2−b′2 look like bx sin(θ)+ay cos(θ)=ab sinθ cosθ by suitable choices of a', b' and ϕ. But note that this must work keeping a' and b' fixed while ϕ is allowed to vary as a function of θ.utkarshakash said:Do you want me to compare the two lines?
haruspex said:Yes. You want to make a′x sec(ϕ)−b′y cosec(ϕ)=a′2−b′2 look like bx sin(θ)+ay cos(θ)=ab sinθ cosθ by suitable choices of a', b' and ϕ. But note that this must work keeping a' and b' fixed while ϕ is allowed to vary as a function of θ.
OK, nearly there. Now you must set a' and b' so that all values of θ and α can occur. (Otherwise there will be a gap in one of the ellipses.)utkarshakash said:OK I did exactly what you said and got the following relations after comparison
[itex] cos\alpha = \dfrac{aa'cos\theta}{a'^2 - b'^2} \\
sin \alpha = \dfrac{-bb'sin\theta}{a'^2 - b'^2} \\
tan \alpha = \dfrac{-bb' tan\theta}{aa'}
[/itex]
A fixed concentric ellipse is a geometric shape that consists of two axes (major and minor) that intersect at a fixed point, known as the center. The shape of the ellipse remains constant no matter where it is located on the plane.
The normal for a fixed concentric ellipse is a line that is perpendicular to the tangent at any point on the ellipse. This means that the normal intersects the tangent at a 90-degree angle.
The normal to a fixed concentric ellipse can be calculated using the derivative of the ellipse's equation. The derivative at any given point on the ellipse will give the slope of the tangent, and the negative reciprocal of this slope will give the slope of the normal.
The normal to a fixed concentric ellipse can be used to determine the curvature of the ellipse at any given point. It is also important in many engineering applications, such as in the design of gears and pulleys.
The eccentricity of an ellipse, which is a measure of how elongated or circular it is, affects the shape and orientation of the normal. As the eccentricity increases, the normal approaches the minor axis and becomes more parallel to it.