Question

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

• A. $-1<m<1$
• B. $m<-1$ or $m>1$
• C. $mϵR$
• D. None of the above

Answer 1

The correct option is D None of the above

Let $y=Y,x-1=X$
Then, the equation becomes $Y^2=4X$.
So, the focus $=(2,0)$
Any line through the focus is $y=m(x-2)$
On solving this with $y^2-4(x-10$
$m^2(x-2{)}^2=4(x-1)$
$\Rightarrow m^2{x}^2-4(m^2+1)x+4(m^2+1)=0$
If $m\ne 0,D=16(m^2+1{)}^2-16m^2(m^2+1)$
$=16(m^2+1)>0$, for all $m$
But, if $m=0$, then $x$ does not have two real distinct values.
So, $mϵR$ except $m=0$
$\therefore mϵR-\left\{0\right\}$.