Question

Assertion :The equation $x^2+2\lambda xy+y^2-2x-2y+2=0$ represent an ellipse if $\lambda \in (-1,1)$. Reason: The general equation of second degree given by $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represent an ellipse if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2\ne 0$ & $h^2<ab$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

The equation $x^2+2\lambda xy+y^2-2x-2y+2=0$ represents ellipse if $-1<\lambda <1$

The general equation of second degree is $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$

It represents an ellipse if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2\ne 0$ and $h^2<ab$

In given equation , $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=2+2\lambda -1-1-2{\lambda }^2\ne 0$ and ${\lambda }^2<1$

$\Rightarrow -1<\lambda <1$

Therefore assertion is correct and reason is the correct explanantion of assertion

Therefore option $A$ is correct