Question

Find the coordinates of the point on the curve $4y = {x^2}$ which are nearest to the point $\left( {0,5} \right)$

Answer 1

Hint:
Here, we will try finding the distance between the two points using the distance formula. Then differentiating the equation formed will help us to find the minimum or the nearest distance between these two pints. Hence, substituting the values in the given equation will give us the required coordinates of the nearest point.

Formula Used:
We will use the following formulas:
1) Distance formula: $D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $
2) $\dfrac{{dy}}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$

Complete step by step solution:
The equation of the curve is $4y = {x^2}$.
Let $\left( {x,y} \right)$ be any point on the curve which is nearest to the point $\left( {0,5} \right)$
By substituting the above points in the formula $D = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} $, we get
Distance, $D = \sqrt {{{\left( {x - 0} \right)}^2} + {{\left( {y - 5} \right)}^2}} $
$ \Rightarrow D = \sqrt {{x^2} + {{\left( {y - 5} \right)}^2}} $
But, according to the equation of the curve, $4y = {x^2}$
Hence, substituting this value in the above equation, we get,
$ \Rightarrow D = \sqrt {4y + {{\left( {y - 5} \right)}^2}} $
Squaring both sides, we get
$ \Rightarrow Z = {D^2} = 4y + {\left( {y - 5} \right)^2}$
Now, differentiate both sides with respect to $y$ using the formula $\dfrac{{dy}}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$, we get
$ \Rightarrow Z\prime = 4 + 2\left( {y - 5} \right)$
Now, we will equate the derivative to 0. Therefore, we get
$ \Rightarrow Z\prime = 4 + 2\left( {y - 5} \right) = 0$
$ \Rightarrow 4 + 2y - 10 = 0$
Subtracting the like terms, we get
$ \Rightarrow 2y - 6 = 0$
Adding 6 on both sides, we get
$ \Rightarrow 2y = 6$
Dividing both sides by 2, we get
$ \Rightarrow y = 3$
We will now substitute this value in the equation of the curve because the point $\left( {x,y} \right)$ satisfies the equation of the curve as it lies on it.
Substituting $y = 3$ in the equation $4y = {x^2}$, we get
${x^2} = 4 \times 3 = 12$
Taking square root on both sides
$ \Rightarrow x = \pm \sqrt {12} = \pm 2\sqrt 3 $
Also $Z\prime = Z > 0$, hence the point obtained is minima.

Therefore, the required nearest point to the point $\left( {0,5} \right)$ is $\left( { \pm 2\sqrt 3 ,3} \right)$.
Hence, this is the required answer.

Note:
A curve is a continuous and smooth line without any sharp turns. It bends and changes its direction at least once and it is a type of line which gradually deviates from being a straight line for some or all of its length. Any point lying on a curve satisfies its equation. In order to find the nearest point to any given point, we try to use minima by differentiating the equation formed. Also, distance formula is a numerical measurement of how far apart points are. Hence, using this will helps us to find the required shortest distance.