Question

The length of the normal chord to the parabola $$y^2=4x$$ which subtends a right angle at the vertex is_____
A. $$6\sqrt{3}$$
B. $$3\sqrt{3}$$
C. $$2$$
D. $$1$$

Answer 1

Hint: A line segment passing through any two points on the parabola is known as a chord, and the chord which is perpendicular to the tangent of the parabola at the point of intersection is known as a normal chord.

Complete step by step answer:

We are assuming CB as the chord to the parabola $$y^2=4x$$ where $$a=1$$
Let $$\left(a{t}^2,2a{t}_1\right)$$$$\left(a{t_2}^2,2a{t}_2\right)$$ be the coordinates of C and B respectively. So, the equation becomes at point C,
$$y-2t^1={\displaystyle \frac{-2t}{2}}\left(x-t^2\right)$$
$$\Rightarrow y-2t^1=-t\left(x-t^2\right)$$.................(equation 1)
Therefore, (slope of CA) (slope of AB)$$=-1$$

As its given in the question, that the normal chord subtends a right angle at the vector
$$\left({\displaystyle \frac{\left(2t-0\right)}{\left(t^2-0\right)}}\right)\left({\displaystyle \frac{\left(2t_2-0\right)}{\left(t_2^2-0\right)}}\right)=-1$$
$$\Rightarrow \left({\displaystyle \frac{2t}{t^2}}\right)\left({\displaystyle \frac{2t_2}{t_2^2}}\right)=-1$$
$$\Rightarrow t_1{t}_2=-4$$..................( equation 2)
From equation 2,
$$-t_1\left(t_2+t_1\right)=2$$
$$\Rightarrow 4-t_1^2=2$$....................(equation 3)
$$\Rightarrow t_1^{}=\sqrt{2}$$

Substituting $$t_1$$ in 3 equation = $$t_2={\displaystyle \frac{-4}{\sqrt{2}}}$$
$$\Rightarrow {\displaystyle \frac{-4\sqrt{2}}{2}}\Rightarrow -2\sqrt{2}$$
Coordinates of C= $$\left(2,2\sqrt{2}\right)$$
Coordinates of B=$$\left(8,-4\sqrt{2}\right)$$
Therefore the length of the normal chord is, here we are using the distance formula we can get the distance between the two coordinates C and B respectively.
$$\begin{gathered} \sqrt{{\left(8-2\right)}^2+{\left(-4\sqrt{2}-2\sqrt{2}\right)}^2} \\ \Rightarrow \sqrt{6^2+{\left(-6\sqrt{2}\right)}^2} \\ \Rightarrow \sqrt{108} \\ \therefore 6\sqrt{3} \end{gathered}$$
In the end we came to a conclusion that the answer is $$6\sqrt{3}$$ units.

So, option A is the correct option.

Note:In the above solution, distance formula has been used to calculate the distance between two coordinates given. It is an application of the Pythagorean theorem.Interestingly, a lot of people don't actually memorize this formula. Instead, they set up a right triangle, and use the Pythagorean theorem whenever they want to find the distance between two points.