Question

Q.Choose the correct answer.
Q. The equation 2x2+7y2-4xy+3y-1=0 represents
a. A pair of straight lines
b. Ellipse
c. Parabola
d. Hyperbola

Answer 1

I'll use invariants.

$2x^2+7y^2-4xy+3y-1=0 \\   \\ a_{11} = 2, a_{22} = 7, a_{12} = -2, a_{13} = 0, a_{23} = 1.5, a_{33} = -1 \\   \\I_2 = \begin{vmatrix} 2 & -2 \\ -2 & 7 \end{vmatrix} = 10 > 0 \\ \\I_3 = \begin{vmatrix} 2 & -2 & 0 \\ -2 & 7 & 1.5\\ 0 & 1.5 & -1 \end{vmatrix} = -14 - 2*2.25 + 4 = -14.5 \\ \\ I_1 = tr\begin{pmatrix} 2 & -2 \\ -2 & 7 \end{pmatrix} = 9 \\ \\ I_2 > 0,    I_1I_3 < 0 \implies Ellipse$

Answer: b