Question

The equation $\lambda x^2+4xy+y^2+\lambda x+3y+2=0$ represents

• A. a parabola if $\lambda =4$
• B. an ellipse if $\lambda >4$
• C. a hyperbola if $\lambda <4$
• D. a pair of straight lines if $\lambda =2$

Answer 1

Answer: (A), (B), (C)

a hyperbola if $\lambda <4$

Comparing the given equation with standard conic equation
$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0,$ we get
$a=\lambda =2g,h=2,b=1,f=3/2,c=2$
Now, $\Delta =abc+2fgh-a{f}^2-b{g}^2-c{h}^2$
$\Rightarrow \Delta =\frac{1}{4}(-{\lambda }^2+11\lambda -32)$
Hence $\Delta \ne 0,\forall \lambda \in R$

Now, for conic to be parabola,
$2^2=1\cdot \lambda \Rightarrow \lambda =4$
For conic to be ellipse, $\lambda >4$
and for conic to be hyperbola, $\lambda <4$