Answer

Verified

411.6k+ views

**Hint:**Assume the variable point on the given curve. Now find its distance from the origin. By using the equation of the curve, simplify the distance of the variable point from the origin. Now differentiate the equation and equate it to $0$ to get the minimum distance between the variable point and the origin.

**Complete step-by-step answer:**

Let us assume the point $(h,k)$ such that it lies on the curve $y = {x^2} - 4$. Since $(h,k)$ lies on the curve, therefore replacing $x$ by $h$ and $y$ by $k$, we get

$k = {h^2} - 4 - - - - - (1)$

Distance between the points $({x_1},{y_1})$ and $({x_2},{y_2})$

Distance formula $ = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_2} - {y_1})}^2}} $

Distance between $(h,k)$ and $(0,0)$

$D$$ = \sqrt {{{(h - 0)}^2} + {{(k - 0)}^2}} $

${D^2} = {h^2} + {k^2}$

From (1), $k = {h^2} - 4$

${h^2} = k + 4$

${D^2} = {k^2} + k + 4$

As stated in the question, we have to find the minimum distance between the point on the curve and the origin.

Hence, for finding the minimum distance we need to differentiate the equation and equate it to $0$.

The value of $k$ for which the equation will satisfy will be the $y$ coordinate of that point.

By differentiating the equation,

$2k + 1 = 0$

$k = \dfrac{{ - 1}}{2}$

Substituting in (1),

$\dfrac{{ - 1}}{2} = {h^2} - 4$

$h = \pm \sqrt {\dfrac{7}{2}} $

$x - $coordinate of the point is $ \pm \sqrt {\dfrac{7}{2}} $

$y - $coordinate of the point is $ - \dfrac{1}{2}$

Therefore minimum distance is given by substituting coordinates value in equation ${D^2} = {h^2} + {k^2}$ we get,

$ = \sqrt {\dfrac{7}{2} + \dfrac{1}{4}} = \dfrac{{\sqrt {15} }}{2}$

**So, the correct answer is “Option A”.**

**Note:**An important step in this question is the formation of the equation (1).Students should remember that for finding maximum or minimum point we have to differentiate the equation and equate it to 0.And also should remember the distance between two points formula i.e $ = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_2} - {y_1})}^2}} $ for solving these types of questions.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail