Question

The length of the portion of the normal at $\left( {1,1} \right)$ to ${x^2} = y$ intercepted between the axes is
A. $\dfrac{{5\sqrt 3 }}{2}$
B. $\dfrac{{3\sqrt 5 }}{2}$
C. $\dfrac{{5\sqrt 3 }}{4}$
D. $\dfrac{{3\sqrt 5 }}{4}$

Answer 1

We will first find the slope of the normal to the curve ${x^2} = y$. Then, form the equation of the line normal to the given curve and pass through $\left( {1,1} \right)$. Next, find the coordinates of $x$ and $y$ intercepts. Hence, apply the distance formula to find the distance between intercepts.

Complete step-by-step answer:
We are given that the equation of the curve is ${x^2} = y$.
We will find the coordinates of normal at $\left( {1,1} \right)$
We will find $\dfrac{{dy}}{{dx}}$ at point $\left( {1,1} \right)$ to find the normal.
Now, we know that $\dfrac{{d\left( {{x^n}} \right)}}{{dx}} = n{x^{n - 1}}$
Then,
$\dfrac{{d\left( {{x^2}} \right)}}{{dx}} = 2x$
That is , $y' = 2x$
Hence the slope of the normal at $\left( {1,1} \right)$ is $\dfrac{{ - 1}}{2}$
Then, the equation of line passing through $\left( {1,1} \right)$
$\left( {y - 1} \right) = - \dfrac{1}{2}\left( {x - 1} \right) \\\ \Rightarrow 2y - 2 = - x + 1 \\\ \Rightarrow 2y + x = 3 \\\$
We will now find $x$ intercept by putting $y$ equals to 0 in the above equation.
$x = 3$
Similarly, we will now find $y$ intercept by putting $x$ equals to 0 in the above equation.
$2y = 3 \\\ y = \dfrac{3}{2} \\\$
Hence, the coordinates of $x$ axis are $\left( {3,0} \right)$ and coordinates of $y$ axis are $\left( {0,\dfrac{3}{2}} \right)$.
We have to find the distance between $\left( {3,0} \right)$ and $\left( {0,\dfrac{3}{2}} \right)$
Also, if $\left( {p,q} \right)$ and $\left( {m,n} \right)$ are two points, then the distance between them is given by $\sqrt {{{\left( {p - m} \right)}^2} + {{\left( {q - n} \right)}^2}}$
Therefore, the distance between $\left( {3,0} \right)$ and $\left( {0,\dfrac{3}{2}} \right)$ is
$\sqrt {{{\left( {3 - 0} \right)}^2} + {{\left( {0 - \dfrac{3}{2}} \right)}^2}} = \sqrt {9 + \dfrac{9}{4}} = \dfrac{{\sqrt {45} }}{2} = \dfrac{{3\sqrt 5 }}{2}$
Hence, option B is correct.

Note: If a line is passing through $\left( {{x_1},{y_1}} \right)$ and the slope of the line is $m$, then the equation of the line is given as $y - {y_1} = m\left( {x - {x_1}} \right)$. Also, the product of slope of perpendicular lines is equal to $- 1$. $x$intercept is the distance from the origin to the point where the line cuts the $x$ axis.