common equation of conic sections ax^2 +bxy+cy^2 +dx+ey+f=0 if b≠0 we rotate tan 2α =(b/(a−c)) [if a=c ⇒ α=45°] { ((x=x′cos α −y′sin α)),((y=x′sin α +y′cos α)) :} we now have [using x, y again instead of x′, y′] Ax^2 +Cy^2 +Dx+Ey+F=0 now complete the squares A(x+(D/(2A)))^2 +C(y+(E/(2C)))^2 +(F−(D^2 /(4A^2 ))−(E^2 /(4C^2 )))=0 { ((x=x′−(D/(2A)))),((y=y′−(E/(2C)))) :} we now have [using x, y again instead of x′, y′] one of these { ((Ax^2 +By^2 +C=0)),((Ax+By^2 +C=0)),((Ax^2 +By+C=0)) :} a, c, A, C, A, B, C ≠0 in all other cases use your brain to interprete which kind of curve (if any) we have


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Question Number 78246 by MJS last updated on 15/Jan/20
$common equation of conic sections ax^2 +bxy+cy^2 +dx+ey+f=0 if b≠0 we rotate tan 2α =(b/(a−c)) [if a=c ⇒ α=45°] { ((x=x′cos α −y′sin α)),((y=x′sin α +y′cos α)) :} we now have [using x, y again instead of x′, y′] Ax^2 +Cy^2 +Dx+Ey+F=0 now complete the squares A(x+(D/(2A)))^2 +C(y+(E/(2C)))^2 +(F−(D^2 /(4A^2 ))−(E^2 /(4C^2 )))=0 { ((x=x′−(D/(2A)))),((y=y′−(E/(2C)))) :} we now have [using x, y again instead of x′, y′] one of these { ((Ax^2 +By^2 +C=0)),((Ax+By^2 +C=0)),((Ax^2 +By+C=0)) :} a, c, A, C, A, B, C ≠0 in all other cases use your brain to interprete which kind of curve (if any) we have$
$common equation of conic sections$ ${a x}^{2} + b x y + {c y}^{2} + d x + e y + f = 0$ $if b \neq 0 we rotate$ $\tan 2 α = \frac{b}{a - c} [if a = c \Rightarrow α = 45 °]$ ${\begin{cases} x = x^{'} \cos α - y^{'} \sin α \\ y = x^{'} \sin α + y^{'} \cos α \end{cases}$ $we now have [using x, y again instead of x^{'}, y^{'}]$ ${A x}^{2} + {C y}^{2} + D x + E y + F = 0$ $now complete the squares$ $A {(x + \frac{D}{2 A})}^{2} + C {(y + \frac{E}{2 C})}^{2} + (F - \frac{D^{2}}{4 A^{2}} - \frac{E^{2}}{4 C^{2}}) = 0$ ${\begin{cases} x = x^{'} - \frac{D}{2 A} \\ y = y^{'} - \frac{E}{2 C} \end{cases}$ $we now have [using x, y again instead of x^{'}, y^{'}]$ $one of these$ ${\begin{cases} A x^{2} + B y^{2} + C = 0 \\ A x + B y^{2} + C = 0 \\ A x^{2} + B y + C = 0 \end{cases}$ $a, c, A, C, A, B, C \neq 0$ $in all other cases use your brain to interprete$ $which kind of curve (if any) we have$
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