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In the context of rotating axes for a hyperbola, what angle should be chosen to align the new axes with the transverse and conjugate axes of the hyperbola?
90 degrees
the angle that makes the transverse axis vertical
any arbitrary angle
For a hyperbola with a horizontal transverse axis, what should be the value of the angle of rotation to align the new axes with the transverse and conjugate axes of the hyperbola?
0 degrees
45 degrees
What is the key goal of rotating the axes for a hyperbola?
To make the hyperbola's center at the origin
To change the focus and directrix of the hyperbola
To simplify the equation of the hyperbola and align it with the new axes
After performing a rotation of axes, the new equation for a hyperbola has a term x y'. What does this term indicate?
The new axes are not orthogonal
The hyperbola is now a parabola
The hyperbola is inclined with respect to the original axes
During a rotation of axes for a hyperbola, which term remains unchanged in the equation?
The squared terms involving x and y
the coefficients of the squared terms
The constant term
If the transverse axis of the original hyperbola is aligned with the y-axis, what angle of rotation should be used to align the new axes with the transverse and conjugate axes of the hyperbola?
When performing a rotation of axes for a hyperbola, what term might appear in the simplified equation that wasn't present in the original equation?
the term involving the original coordinates (x, y)
the term involving the new coordinates (x', y')
When performing a rotation of axes for a hyperbola, what happens to the shape of the hyperbola?
It becomes a straight line
It becomes a circle
Its shape remains the same
What is the primary objective of performing a rotation of axes for a hyperbola?
To change the focus and directrix of the hyperbola.
To simplify the equation of the hyperbola and align it with the coordinate axes.
To increase the eccentricity of the hyperbola
When performing a rotation of axes for a hyperbola, what happens to the lengths of the transverse and conjugate axes?
They both increase
They both decrease
The transverse axis increases while the conjugate axis decreases
It is done.