Answer
Verified
419.4k+ views
Hint: We will use the concept of 3D coordinate geometry to solve this question. We know that the minimum distance between any two points is its perpendicular distance. Thus, we will try to find the perpendicular distance between the given plane and point.
Complete step-by-step answer:
We have to find the minimum value of ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$, when the given condition is $ax+by+cz=p$. We will use the concept of 3D coordinate geometry to solve this question. The given equation is the equation of the plane. So, we will find the distance of the point $(x,y,z)$ which is lying somewhere on the plane $ax+by+cz=p$ and the origin.
We have assumed that the point $(x,y,z)$ is perpendicular to origin, this is because the minimum distance between any two points is given by the perpendicular distance and also we have to find the minimum value of ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$. We know that the distance between two points is as,
$d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}$
So, the distance between point $(x,y,z)$ to the origin
is.$d=\sqrt{{{(x-0)}^{2}}+{{(y-0)}^{2}}+{{(z-0)}^{2}}}$.
$\Rightarrow d=\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$
Squaring on both sides, we get
$\Rightarrow {{d}^{2}}={{x}^{2}}+{{y}^{2}}+{{z}^{2}}$
Also, we know that distance between any plane and a point is given as.
${{d}_{\min }}=\left| \dfrac{-p}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right|$, we will use the direct formula of distance between plane and point by squaring, we get
$d_{\min }^{2}=\dfrac{-p}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$
Thus, we get $d_{\min }^{2}=\dfrac{{{p}^{2}}}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}$
$=({{x}^{2}}+{{y}^{2}}+{{z}^{2}})$
Therefore, the minimum value of $({{x}^{2}}+{{y}^{2}}+{{z}^{2}})$ is
$\dfrac{{{p}^{2}}}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}$ and option (B) is the correct answer.
Note: There are many alternate possible methods for solving this particular question but the other methods are slightly complex to solve and the probability of error while solving is higher. But if we follow the this method we will get the minimum value of ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$ directly. Some of the alternate methods are:
1. We can solve this question using the Lagrange multipliers method. In this method, we will find the value of $x,y,z$ individually to get minimum value.
2. We can also use the vector concept or we can name it as the Cauchy Schwarz inequality method. In this method we will assume two vectors $\overrightarrow{A}=(a,b,c)$ and $\overrightarrow{B}=(x,y,z)$, then solve the question to get the minimum value.
Complete step-by-step answer:
We have to find the minimum value of ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$, when the given condition is $ax+by+cz=p$. We will use the concept of 3D coordinate geometry to solve this question. The given equation is the equation of the plane. So, we will find the distance of the point $(x,y,z)$ which is lying somewhere on the plane $ax+by+cz=p$ and the origin.
We have assumed that the point $(x,y,z)$ is perpendicular to origin, this is because the minimum distance between any two points is given by the perpendicular distance and also we have to find the minimum value of ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$. We know that the distance between two points is as,
$d=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}$
So, the distance between point $(x,y,z)$ to the origin
is.$d=\sqrt{{{(x-0)}^{2}}+{{(y-0)}^{2}}+{{(z-0)}^{2}}}$.
$\Rightarrow d=\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$
Squaring on both sides, we get
$\Rightarrow {{d}^{2}}={{x}^{2}}+{{y}^{2}}+{{z}^{2}}$
Also, we know that distance between any plane and a point is given as.
${{d}_{\min }}=\left| \dfrac{-p}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right|$, we will use the direct formula of distance between plane and point by squaring, we get
$d_{\min }^{2}=\dfrac{-p}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}$
Thus, we get $d_{\min }^{2}=\dfrac{{{p}^{2}}}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}$
$=({{x}^{2}}+{{y}^{2}}+{{z}^{2}})$
Therefore, the minimum value of $({{x}^{2}}+{{y}^{2}}+{{z}^{2}})$ is
$\dfrac{{{p}^{2}}}{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}$ and option (B) is the correct answer.
Note: There are many alternate possible methods for solving this particular question but the other methods are slightly complex to solve and the probability of error while solving is higher. But if we follow the this method we will get the minimum value of ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$ directly. Some of the alternate methods are:
1. We can solve this question using the Lagrange multipliers method. In this method, we will find the value of $x,y,z$ individually to get minimum value.
2. We can also use the vector concept or we can name it as the Cauchy Schwarz inequality method. In this method we will assume two vectors $\overrightarrow{A}=(a,b,c)$ and $\overrightarrow{B}=(x,y,z)$, then solve the question to get the minimum value.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Summary of the poem Where the Mind is Without Fear class 8 english CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Write an application to the principal requesting five class 10 english CBSE
What organs are located on the left side of your body class 11 biology CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE