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Question

The equation λx2+4xy+y2+λx+3y+2=0 represents
  1. a parabola if λ=4
  2. an ellipse if λ>4
  3. a hyperbola if λ<4
  4. a pair of straight lines if λ=2


Solution

The correct options are
A a parabola if λ=4
B an ellipse if λ>4
C a hyperbola if λ<4
Comparing the given equation with standard conic equation
ax2+2hxy+by2+2gx+2fy+c=0, we get
a=λ=2g,h=2,b=1,f=3/2,c=2
Now, Δ=abc+2fghaf2bg2ch2
Δ=14(λ2+11λ32)
Hence Δ0,    λR

Now, for conic to be parabola,
22=1λλ=4
For conic to be ellipse, λ>4
and for conic to be hyperbola, λ<4

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