Question

A line $L$ passing through the focus of the parabola $y^2=4(x-1)$ intersects the parabola in two distinct points. If $m$ be the slope of the line $L$ then

• A. $-1<m<1$
• B. $m<-1$ or $m>1$
• C. $m\in R$
• D. none of these

Answer 1

The correct option is D none of these

Let the equation of line be $y=mx+c$

Since, it passes through focus $(2,0)$

Then $c=-2m$

Substitute L in parabola

$(mx-2m{)}^2=4(x-1)$

$m^2{x}^2-4x(m^2+1)+4(1+m^2)=0$

Since, it intersection in two distinct points

Therefore, $b^2-4ac=16(1+m^2{)}^2-16(1+m^2)>0$

$\Rightarrow m\in R$ and $m\ne 0$

Hence, $m\in R-\left\{0\right\}$