Question

Assertion :Pair of straight lines represented by the equation $(2\lambda +7)x^2+2\lambda xy+{\lambda }^2{y}^2=0$ are not perpendicular for any real value of $\lambda$. Reason: If for a non-zero value of $\lambda$, one of the lines represented by the equation $(2\lambda +1)x^2+\lambda xy-({\lambda }^2+1)y^2=0$ is $x+y=0$, the other line is $3x-2y=0$.

• 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 B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Assertion:

$(2\lambda +7)x^2+2\lambda xy+{\lambda }^2{y}^2=0$ represent pair of perpendicular lines if
$2\lambda +7+{\lambda }^2=0\Rightarrow {\lambda }^2+2\lambda +7=0$
As roots are imaginary,
So this pair of line is not perpendicular for any value of $\lambda$
Reason:
For $\lambda =1$
$(2\lambda +1)x^2+\lambda xy-({\lambda }^2+1)y^2=0\Rightarrow {3x}^2+xy-{2y}^2=0$
$\Rightarrow (x+y)(3x-2y)=0\Rightarrow x+y=0$ and $3x-2y=0.$