Answer
Verified
480.9k+ views
Hint- Use a family of parallel planes . Equation of all planes parallel to $ax + by + cz + d = 0$ lies on this family $ax + by + cz + e = 0$.
Now, the equation of all planes parallel to $3x + 2y - 2z + 15 = 0$ lies on the family of planes $3x + 2y - 2z + d = 0$.
But we require a unique plane pass through the point $\left( {3,8,2} \right)$. So, we put the point in the family of planes.
$
3 \times 3 + 2 \times 8 - 2 \times 2 + d = 0 \\
\Rightarrow 9 + 16 - 4 + d = 0 \\
\Rightarrow d = - 21 \\
$
So, the required plane is $3x + 2y - 2z - 21 = 0..........\left( 2 \right)$
The required plane also passes through the given line. So, we find an intersection point.
Equation of line $\dfrac{{x - 1}}{2} = \dfrac{{y - 3}}{4} = \dfrac{{z - 2}}{3} = p$
$ \Rightarrow x = 2p + 1,y = 4p + 3,z = 3p + 2$
These points also satisfy the equation of the plane. So,
$
\Rightarrow 3\left( {2p + 1} \right) + 2(4p + 3) - 2(3p + 2) - 21 = 0 \\
\Rightarrow 6p + 3 + 8p + 6 - 6p - 4 - 21 = 0 \\
\Rightarrow 8p = 16 \\
\Rightarrow p = 2 \\
$
Intersection points of the plane and line are $x = 2 \times 2 + 1 = 5$ ,$y = 4 \times 2 + 3 = 11$ , $z = 3 \times 2 + 2 = 8$ .
Coordinate of intersection $\left( {5,11,8} \right)$ .
Distance of point $\left( {3,8,2} \right)$ from the line $\dfrac{{x - 1}}{2} = \dfrac{{y - 3}}{4} = \dfrac{{z - 2}}{3}$ is equal to distance between $\left( {3,8,2} \right)$ and $\left( {5,11,8} \right)$.
Distance $
= \sqrt {{{\left( {5 - 3} \right)}^2} + {{\left( {11 - 8} \right)}^2} + {{\left( {8 - 2} \right)}^2}} \\
\\
$
$
\Rightarrow \sqrt {4 + 9 + 36} \\
\Rightarrow \sqrt {49} = 7 \\
$
So, the correct option is (d).
Note-Whenever we face such types of problems we use some important points. First find the equation of plane passing through the point and parallel to another plane with the help of the family of planes then find the point of intersection of plane and line then using distance formula we get the required answer.
Now, the equation of all planes parallel to $3x + 2y - 2z + 15 = 0$ lies on the family of planes $3x + 2y - 2z + d = 0$.
But we require a unique plane pass through the point $\left( {3,8,2} \right)$. So, we put the point in the family of planes.
$
3 \times 3 + 2 \times 8 - 2 \times 2 + d = 0 \\
\Rightarrow 9 + 16 - 4 + d = 0 \\
\Rightarrow d = - 21 \\
$
So, the required plane is $3x + 2y - 2z - 21 = 0..........\left( 2 \right)$
The required plane also passes through the given line. So, we find an intersection point.
Equation of line $\dfrac{{x - 1}}{2} = \dfrac{{y - 3}}{4} = \dfrac{{z - 2}}{3} = p$
$ \Rightarrow x = 2p + 1,y = 4p + 3,z = 3p + 2$
These points also satisfy the equation of the plane. So,
$
\Rightarrow 3\left( {2p + 1} \right) + 2(4p + 3) - 2(3p + 2) - 21 = 0 \\
\Rightarrow 6p + 3 + 8p + 6 - 6p - 4 - 21 = 0 \\
\Rightarrow 8p = 16 \\
\Rightarrow p = 2 \\
$
Intersection points of the plane and line are $x = 2 \times 2 + 1 = 5$ ,$y = 4 \times 2 + 3 = 11$ , $z = 3 \times 2 + 2 = 8$ .
Coordinate of intersection $\left( {5,11,8} \right)$ .
Distance of point $\left( {3,8,2} \right)$ from the line $\dfrac{{x - 1}}{2} = \dfrac{{y - 3}}{4} = \dfrac{{z - 2}}{3}$ is equal to distance between $\left( {3,8,2} \right)$ and $\left( {5,11,8} \right)$.
Distance $
= \sqrt {{{\left( {5 - 3} \right)}^2} + {{\left( {11 - 8} \right)}^2} + {{\left( {8 - 2} \right)}^2}} \\
\\
$
$
\Rightarrow \sqrt {4 + 9 + 36} \\
\Rightarrow \sqrt {49} = 7 \\
$
So, the correct option is (d).
Note-Whenever we face such types of problems we use some important points. First find the equation of plane passing through the point and parallel to another plane with the help of the family of planes then find the point of intersection of plane and line then using distance formula we get the required answer.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
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
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
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
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE