I have equation of parabola $(ax+by)^2+2fy=0$ and I have to find axis of this parabola so I made the substitution $X = ax +by$ and $Y= \frac{x}{a}-\frac{y}{b}$ and then solving by these substitution I m getting axis of symmetry $$ax+by = \frac{-bf}{a^2+b^2}$$ this is correct answer according to book. but the problem is I have made random substitution to bring the answer and I do not know why this substitution works and I also want to know how can we approach to find axis of symmetry for more general case $$(ax+by)^2+2gx+2fy+c=0$$

  • $\begingroup$ Rotate the axes by an angle $\theta$ to get rid of $xy$ term $\endgroup$ – Ekaveera Kumar Sharma Dec 31 '15 at 14:54
  • $\begingroup$ @EkaveeraKumarSharma how to determine this $\theta$ $\endgroup$ – TIWARI Dec 31 '15 at 14:58
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    $\begingroup$ you rotate and find $\theta$ such that $xy$ term in the resulting equation is zero. $\endgroup$ – Ekaveera Kumar Sharma Dec 31 '15 at 15:00
  • $\begingroup$ my substitution is rotating the parabola at correct angle in addition with scaling the axis which doesn't affect the equation of axis of symmetry. $\endgroup$ – TIWARI Jan 1 '16 at 6:29

This page tells you how to find the angle of rotation. As the description indicates, the same technique can be applied to any conic section (i.e. to any second degree equation), not just an equation representing a parabola.

As one of the comments suggested, the basic idea is to write your equation using a coordinate system that is rotated by some (as yet unknown) angle $\theta$. Then you choose the value of $\theta$ that makes the coefficient of the $xy$ term zero.


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