Question

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

• A. $m\in R-\left\{0\right\}$
• B. $-1<m<1$
• C. $m<-1$ or $m>1$
• D. None of these

Answer 1

The correct option is A $m\in R-\left\{0\right\}$

The focus of the parabola $y^2=4(x-1)$ is $(2,0).$
Any line through the focus is
$(y-0)=m(x-2),i.e.,y=m(x-2)$
It will meet the given parabola if
$m^2(x-2{)}^2=4(x-1)$
or $m^2{x}^2-4(m^2+1)x+4(m^2+1)=0$
If $m\ne 0,$ then
discriminant $=16(m^2+1{)}^2-16m^2(m^2+1)$
$=16(m^2+1)>0\forall m\in R$
But if $m=0,$ then $x$ does not have two real and distinct values.
$\therefore m\in R-\left\{0\right\}$