Q. Equation of circle- $2x^2+ \lambda xy+2y^2+( \lambda -4)x+6y-5=0$ find area of the circle.
Attempt- For converting the equation from second degree to first degree $\lambda xy=0$.
Thus, $\lambda =0$ and-
$$(\lambda -4)x = 2gx$$ $$ 6y=2fy$$ $$c=-5$$ $$g=-2, f=3, c=-5$$
Radius of circle = $\sqrt{4+9+5}=\sqrt{18}$
Area of circle= $ \pi *18$
But the answer is $\frac {23}{4} * \pi$
Complete squares after putting $\;\lambda xy=0\implies \lambda =0\;$:
$$0=2x^2+2y^2-4x+6y-5=2(x-1)^2-2+2\left(y-\frac32\right)^2-\frac92-5\implies$$
$$\implies2(x-1)^2+2\left(y-\frac32\right)^2=\frac{23}2\implies(x-1)^2+\left(y-\frac32\right)^2=\frac{23}4$$
and we have a circle of radius $\;\sqrt{\frac{23}4}\;$ , so its area is
$$\pi\sqrt{\frac{23}4}^2=\frac{23\pi}4$$