Answer
Verified
455.7k+ 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$$ = \sqrt {{{(h - 0)}^2} + {{(k - 0)}^2}} $
${D^2} = {h^2} + {k^2}$
From (1), $k = {h^2} - 4$
${h^2} = k + 4$
${D^2} = {k^2} + k + 4$
As stated in the question, we have to find the minimum distance between the point on the curve and the origin.
Hence, for finding the minimum distance we need to differentiate the equation and equate it to $0$.
The value of $k$ for which the equation will satisfy will be the $y$ coordinate of that point.
By differentiating the equation,
$2k + 1 = 0$
$k = \dfrac{{ - 1}}{2}$
Substituting in (1),
$\dfrac{{ - 1}}{2} = {h^2} - 4$
$h = \pm \sqrt {\dfrac{7}{2}} $
$x - $coordinate of the point is $ \pm \sqrt {\dfrac{7}{2}} $
$y - $coordinate of the point is $ - \dfrac{1}{2}$
Therefore minimum distance is given by substituting coordinates value in equation ${D^2} = {h^2} + {k^2}$ we get,
$ = \sqrt {\dfrac{7}{2} + \dfrac{1}{4}} = \dfrac{{\sqrt {15} }}{2}$
So, the correct answer is “Option A”.
Note:An important step in this question is the formation of the equation (1).Students should remember that for finding maximum or minimum point we have to differentiate the equation and equate it to 0.And also should remember the distance between two points formula i.e $ = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_2} - {y_1})}^2}} $ for solving these types of questions.
Complete step-by-step answer:
Let us assume the point $(h,k)$ such that it lies on the curve $y = {x^2} - 4$. Since $(h,k)$ lies on the curve, therefore replacing $x$ by $h$ and $y$ by $k$, we get
$k = {h^2} - 4 - - - - - (1)$
Distance between the points $({x_1},{y_1})$ and $({x_2},{y_2})$
Distance formula $ = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_2} - {y_1})}^2}} $
Distance between $(h,k)$ and $(0,0)$
$D$$ = \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
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
What percentage of the solar systems mass is found class 8 physics CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Why is there a time difference of about 5 hours between class 10 social science CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE