Question

The set of points on the axis of the parabola $y^2-2y-4x+5=0$ from which all the three normals to the parabola are real is :

• A. $\left\{\right(x,1);x\ge 3\}$
• B. $\left\{\right(x,-1);x\ge 1\}$
• C. $\left\{\right(x,3);x\ge 1\}$
• D. $\left\{\right(x,-3);x\ge 3\}$

Answer 1

The correct option is A $\left\{\right(x,1);x\ge 3\}$

$y^2-2y-4x+5=0\Rightarrow (y-1{)}^2=4(x-1)$
Axis of parabola is
$y-1=0\Rightarrow y=1$
Equation of normal is
$(y-1)=m(x-1)-2m-m^3$

Let $(h,1)$ be any point on its axis, then
$0=m(h-1)-2m-m^3\Rightarrow m=0,m^2=h-3$
For three normals to exist,
$\Rightarrow h\ge 3$

So, $x\ge 3\text{and}y=1$