Question

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

• A. $\left\{\left(x , 1\right) : x > 3\right\}$
• B. $\left\{\left(x , - 1\right) : x \geq 1\right\}$
• C. $\left\{\left(x , 3\right) : x \geq 1\right\}$
• D. $\left\{\left(x , - 3\right) : x \geq 3\right\}$

Answer 1

Answer: (A)

Given equation of the parabola can be rewritten as
$\left(y - 1\right)^{2}=4\left(x - 1\right)$
Axis of the parabola is $y=1$ ,
equation of normal is
$\left(y - 1\right)=m\left(x - 1\right)-2m-m^{3}$
Let $\left(h , \, 1\right)$ is a point on its axis, then
$0=m\left(h - 1\right)-2m-m^{3}$
$\Rightarrow m^{2}=h-3$
$\Rightarrow h>3$ for real & distinct values of $m$ .