Find the equation of the plane that bisects the line joining $\left( {1,2,3} \right)$ and $\left( {3,4,5} \right)$, and is at the right angle to the line.
Last updated date: 25th Mar 2023
•
Total views: 309k
•
Views today: 4.89k
Answer
309k+ views
Hint: Plane is perpendicular to the line, so the direction ratios of the plane is equal to the direction ratios of the line.
Line passing through the points $\left( {1,2,3} \right)$ and $\left( {3,4,5} \right)$
So the midpoint of the line be$\left( {\dfrac{{1 + 3}}{2},\dfrac{{2 + 4}}{2},\dfrac{{3 + 5}}{2}} \right)$
$ \Rightarrow \left( {\dfrac{4}{2},\dfrac{6}{2},\dfrac{8}{2}} \right) = \left( {2,3,4} \right)$
The direction ratios of the line be $\left( {\left( {3 - 1} \right),\left( {4 - 2} \right),\left( {5 - 3} \right)} \right) = \left( {2,2,2} \right)$
Now, it is given that the plane bisects the line so the plane is passing through midpoint of the line and plane is also perpendicular to the line
Therefore the direction ratios of the plane is equal to the direction ratios of the line.
Let the direction ration of the plane be $\left( {l,m,n} \right)$
$ \Rightarrow \left( {l,m,n} \right) = \left( {2,2,2} \right)$
So the equation of the plane passing through midpoint of the plane $\left( {2,3,4} \right)$ having direction ratios $\left( {l,m,n} \right)$ be
$l\left( {x - 2} \right) + m\left( {y - 3} \right) + n\left( {z - 4} \right) = 0$
Now substitute the value of $\left( {l,m,n} \right)$
$
2\left( {x - 2} \right) + 2\left( {y - 3} \right) + 2\left( {z - 4} \right) = 0 \\
\Rightarrow x + y + z - 9 = 0 \\
\Rightarrow x + y + z = 9 \\
$
So, this is the required equation of the plane.
Note: In such types of questions first find out the midpoint of the line then find out the direction ratios of the line, then remember the key concept that if the plane is perpendicular to the line then the direction ratios of the plane is equal to the direction ratios of the line, then write the equation passing through the midpoint having direction ratios same as line we will get the required answer.
Line passing through the points $\left( {1,2,3} \right)$ and $\left( {3,4,5} \right)$
So the midpoint of the line be$\left( {\dfrac{{1 + 3}}{2},\dfrac{{2 + 4}}{2},\dfrac{{3 + 5}}{2}} \right)$
$ \Rightarrow \left( {\dfrac{4}{2},\dfrac{6}{2},\dfrac{8}{2}} \right) = \left( {2,3,4} \right)$
The direction ratios of the line be $\left( {\left( {3 - 1} \right),\left( {4 - 2} \right),\left( {5 - 3} \right)} \right) = \left( {2,2,2} \right)$
Now, it is given that the plane bisects the line so the plane is passing through midpoint of the line and plane is also perpendicular to the line
Therefore the direction ratios of the plane is equal to the direction ratios of the line.
Let the direction ration of the plane be $\left( {l,m,n} \right)$
$ \Rightarrow \left( {l,m,n} \right) = \left( {2,2,2} \right)$
So the equation of the plane passing through midpoint of the plane $\left( {2,3,4} \right)$ having direction ratios $\left( {l,m,n} \right)$ be
$l\left( {x - 2} \right) + m\left( {y - 3} \right) + n\left( {z - 4} \right) = 0$
Now substitute the value of $\left( {l,m,n} \right)$
$
2\left( {x - 2} \right) + 2\left( {y - 3} \right) + 2\left( {z - 4} \right) = 0 \\
\Rightarrow x + y + z - 9 = 0 \\
\Rightarrow x + y + z = 9 \\
$
So, this is the required equation of the plane.
Note: In such types of questions first find out the midpoint of the line then find out the direction ratios of the line, then remember the key concept that if the plane is perpendicular to the line then the direction ratios of the plane is equal to the direction ratios of the line, then write the equation passing through the midpoint having direction ratios same as line we will get the required answer.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
