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 $\left\{\right(x,1):x\ge 3\}$

The given parabola can be rewritten as:
$(y-1{)}^2=4(x-1)$
Let, $y-1=Y$ and $x-1=X$
$Y^2=4X$
Now, let $P(h,0)$ be a point on the axis in the new coordinate frame.
The equation of a normal can be written as:
$Y=mX-2m-m^3$
$P$ lies on the normal.
$\Rightarrow m(h-2-m^2)=0$
$\Rightarrow m=0$ or $m^2=h-2$
For three distinct normals to exist,
$h-2\ge 0$
$\Rightarrow x-3\ge 0$ ($x=h-1$)
$\Rightarrow x\ge 3$
$y=1$
Hence, option A is correct.