Question

The straight line $y=2x+\lambda$ does not meet the parabola $y^2=2x$, if

• A. $\lambda <\frac{1}{4}$
• B. $\lambda >\frac{1}{4}$
• C. $\lambda =4$
• D. $\lambda =1$

Answer 1

Answer: (B)

$\lambda >\frac{1}{4}$

$Line:y=2x+\lambda .......\left(i\right)$

Parabola: $y^2=2x......\left(ii\right)$

Solving $\left(i\right)\&\left(ii\right)$ we get, $(2x+\lambda {)}^2=2x$

$\Rightarrow 4x^2+4x\lambda +{\lambda }^2=2x$

$\Rightarrow 4x^2+2x(2\lambda -1)+{\lambda }^2=0$

Line does not meet the parabola

$\Rightarrow D<0$

$\Rightarrow 4(2\lambda -1{)}^2-4(4\left)\right({\lambda }^2)<0$

$\Rightarrow 4[4{\lambda }^2+1-4\lambda -4{\lambda }^2]<0$

$\Rightarrow 1-4\lambda <0$

$\Rightarrow \lambda >\frac{1}{4}$