Question

Identify the type of the conic 4(x-2y+1)^2 +9(2x+y+2)^2 = 25

Answer 1

Given $4(x-2y+1)^2 + 9(2x+y+2)^2$ $= 25$

Divide through by 25, this gives us

$\frac{(x-2y+1)^2}{\frac{25}{4}} + \frac{(2x+y+2)^2}{\frac{25}{9}}$

$=\frac{(x-(2y-1))^2}{(\frac{5}{2})^2} + \frac{((y-(-2x-2))^2}{(\frac{5}{3})^2}=1$

which satisfies the equation of ellipse

$\frac{(x1)^2}{a^2} + \frac{(y1)^2}{b^2}=1$ ,

where $x1, y1$ are new variables.

Therefore the conic is an ellipse.