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#### Pranav

##### Well-known member

- Nov 4, 2013

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**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".

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