# Axis of symmetry of parabola

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 $$(ax+by)^2+2gx+2fy+c=0$$

• Rotate the axes by an angle $\theta$ to get rid of $xy$ term – Ekaveera Kumar Sharma Dec 31 '15 at 14:54
• @EkaveeraKumarSharma how to determine this $\theta$ – TIWARI Dec 31 '15 at 14:58
• you rotate and find $\theta$ such that $xy$ term in the resulting equation is zero. – Ekaveera Kumar Sharma Dec 31 '15 at 15:00
• 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. – TIWARI Jan 1 '16 at 6:29

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.