Question

find the point on the parabola y²=2x os the closest point to the point 1, 4​

Answer 1

$\large\underline{\sf{Solution-}}$

Given equation of Parabola is

$\rm :\longmapsto\: {y}^{2} = 2x - - - (1)$

Let the point P (x, y) be the point on the parabola which is closest to the point Q (1, 4).

So, Distance between P and Q is

$\rm :\longmapsto\:PQ = \sqrt{ {(x - 1)}^{2} + {(y - 4)}^{2} }$

$\rm :\longmapsto\:PQ^{2} = { \bigg(\dfrac{ {y}^{2} }{2} - 1 \bigg)}^{2} + {(y - 4)}^{2}$

$\rm :\longmapsto\:Let \: f(y) = { \bigg(\dfrac{ {y}^{2} }{2} - 1 \bigg)}^{2} + {(y - 4)}^{2}$

$\rm \: \: = \: \dfrac{ {y}^{4} }{4} + 1 - {y}^{2} + {y}^{2} + 16 - 8y$

$\rm \: \: = \: \dfrac{ {y}^{4} }{4} + 17 - 8y$

$\rm \: \: f(y) = \: \dfrac{ {y}^{4} }{4} + 17 - 8y$

Differentiate both sides w. r. t. y, we get

$\rm :\longmapsto\:f'(y) = \dfrac{4 {y}^{3} }{4} - 8$

$\rm :\longmapsto\:f'(y) = {y}^{3} - 8 - - - - - (2)$

For maxima or minima,

$\rm :\longmapsto\:f'(y) =0$

$\rm :\longmapsto\: {y}^{3} - 8 = 0$

$\rm :\longmapsto\: {y}^{3} = 8$

$\bf\implies \:y = 2 - - - - (3)$

On differentiating both sides w. r. t. y equation (2), we get

$\rm :\longmapsto\:f''(y) = {3y}^{2}$

$\rm :\longmapsto\:f''(2) = {3} \times 2^{2}$

$\rm :\longmapsto\:f''(2) = {3} \times 4$

$\rm :\longmapsto\:f''(2) = 12$

$\bf\implies \:f''(2) > 0$

Hence,

$\rm :\longmapsto\:f(y) \: is \: minimum \: when \: y \: = \: 2$

$\bf :\longmapsto\:PQ \: is \: minimum \: when \: y \: = \: 2$

On substituting the value of y = 2, in equation (1), we get

$\rm :\longmapsto\: {2}^{2} = 2x$

$\rm :\longmapsto\: 4 = 2x$

$\bf\implies \:x = 2$

$\bf\implies \:P(2,2) \: is \: point \: on \: {y}^{2} = 2x \: closest \: to \: (1,4)$

Additional Information :-

HOW TO FIND MAXIMUM AND MINIMUM VALUE OF A FUNCTION

• Differentiate the given function, f(x)

• let f'(x) = 0 and find critical points say a.

• Find the second derivative of f(x), say f''(x).

• Apply these critical points in the second derivative.

• The function f (x) is maximum at x = a when f''(a) < 0.

• The function f (x) is minimum at x = a when f''(a) > 0.