Normals from a point to a parabola

[Saitama]

Problem:

STATEMENT-1: Through (h,h+1), there cannot be more than one normal to the parabola $y^2=4x$, if $h<2$.

STATEMENT-2: The point (h,h+1) lies outside the parabola for all $h\neq 1$.

A)Statement-1 is True, Statement-2 is True; Statement 2 is a correct explanation for Statement-1.

B)Statement-1 is True, Statement-2 is True; Statement 2 is NOT a correct explanation for Statement-1.

C)Statement-1 is True, Statement-2 is False.

D)Statement-1 is False, Statement-2 is True.

Attempt:

I figured out that the locus of the given point is $y=x+1$. I found that this equation is a tangent to given parabola, hence Statement-2 is certainly true.

I am unsure about how to proceed for Statement-1. Here's what I think:

The parametric coordinates of the given parabola is $(t^2,2t)$. The equation of normal in terms of $t$ is $y=-tx+2t+t^3$. As this normal passes through $(h,h+1)$, hence,

$$t^3+t(2-h)-(h+1)=0$$

But I am not sure how to proceed from here.

Any help is appreciated. Thanks!

EDIT: The title should be "Normals from a point to parabola". :p

 

[Opalg]

Re: Normals from a point to parabola

The parametric coordinates of the given parabola are $(t^2,2t)$. The equation of normal in terms of $t$ is $y=-tx+2t+t^3$. As this normal passes through $(h,h+1)$, hence,

$$t^3+t(2-h)-(h+1)=0$$

But I am not sure how to proceed from here.

You want to know whether that cubic equation for $t$ has more than one solution. The cubic polynomial $f(t) = t^3+t(2-h)-(h+1)$ has either one real root or three real roots. If it has three roots then its graph must cross the axis three times. It would have to have a local maximum between the first two roots, and a local minimum between the second and third roots. So you could use calculus to find where (if anywhere) the turning points of $f(t)$ occur.

 

[Saitama]

Re: Normals from a point to parabola

Hi Opalg!

You want to know whether that cubic equation for $t$ has more than one solution. The cubic polynomial $f(t) = t^3+t(2-h)-(h+1)$ has either one real root or three real roots. If it has three roots then its graph must cross the axis three times. It would have to have a local maximum between the first two roots, and a local minimum between the second and third roots. So you could use calculus to find where (if anywhere) the turning points of $f(t)$ occur.

The question says that there should be only one normal i.e one real root. For the cubic to have only one real root, $f'(t)>0$ (f(t) must be strictly increasing) $\Rightarrow t^2>h/3$. I don't see what to do with the inequality I have got. :(

 

[Opalg]

Re: Normals from a point to parabola

For the cubic to have only one real root, $f'(t)>0$ (f(t) must be strictly increasing) $\Rightarrow t^2>h/3$. I don't see what to do with the inequality I have got. :(

That's not what I get from differentiating $f(t).$ In fact, $f'(t) = 3t^2 - (h-2).$

 

[Saitama]

Re: Normals from a point to parabola

That's not what I get from differentiating $f(t).$ In fact, $f'(t) = 3t^2 - (h-2).$

Sorry.

But the question still remains, how am I supposed to solve $t^2>(h-2)/3$? I am thinking that if h<2, the inequality always holds true but for any other h, we can always find a t which does not satisfy the inequality and hence, h<2 is the answer. Is this correct?

 

[Opalg]

Re: Normals from a point to parabola

But the question still remains, how am I supposed to solve $t^2>(h-2)/3$? I am thinking that if h<2, the inequality always holds true but for any other h, we can always find a t which does not satisfy the inequality and hence, h<2 is the answer. Is this correct?

STATEMENT-1 is only concerned with what happens when $h<2$. You don't need to worry about what happens for any other $h.$

 

[Saitama]

Re: Normals from a point to parabola

STATEMENT-1 is only concerned with what happens when $h<2$. You don't need to worry about what happens for any other $h.$

Thanks Opalg! :)

I think I will go with B because I don't see how STATEMENT-1 can be deduced from STATEMENT-2.

 

[Opalg]

Re: Normals from a point to parabola

I think I will go with B because I don't see how STATEMENT-1 can be deduced from STATEMENT-2.

Nor do I. (Thinking)