Question

The equation \(\lambda x^2+4xy+y^2+\lambda x+3y+2=0\) represents a parabola, if \(\lambda\) is

• A. 0
• B. 1
• C. 2
• D. 4

Hint:

For \(Ax^2+2Hxy+By^2\), parabola condition is \(AB-H^2=0\). Here \(A=\lambda,B=1,H=2\), so \(\lambda=4\).

Answer 1

The Correct Option is D

Step 1: Identify quadratic form coefficients.
General second degree equation:
\[ Ax^2+2Hxy+By^2+\cdots=0 \] Here:
\[ A=\lambda,\quad 2H=4\Rightarrow H=2,\quad B=1 \] Step 2: Condition for parabola.
For parabola:
\[ AB-H^2=0 \] Step 3: Substitute values.
\[ \lambda(1)-(2)^2=0 \Rightarrow \lambda-4=0 \Rightarrow \lambda=4 \] Final Answer: \[ \boxed{4} \]