3D Geometry 3 Question 22
####22. The plane which bisects the line segment joining the points $(-3,-3,4)$ and $(3,7,6)$ at right angles, passes through which one of the following points?
(2019 Main, 10 Jan II)
(a) $(4,-1,7)$
(b) $(2,1,3)$
(c) $(-2,3,5)$
(d) $(4,1,-2)$
Show Answer
Answer:
(d)
Solution:
- Let the given points be $A(-3,-3,4)$ and $B(3,76)$.
Then, mid-point of line joining $A, B$ is
$ P \frac{-3+3}{2}, \frac{-3+7}{2}, \frac{4+6}{2}=P(0,2,5) $
$\because$ The required plane is perpendicular
bisector of line joining $A, B$, so direction ratios of normal to the plane is proportional to the direction ratios of line joining $A, B$.
So, direction ratios of normal to the plane are $6,10,2$.
$[\because$ DR’s of $A B$ are $3+3,7+3,6-4$, i.e. $6,10,2]$
Now, equation of plane is given by
$ \begin{aligned} a\left(x-x_{1}\right)+b\left(y-y_{1}\right)+c\left(z-z_{1}\right) & =0 \\ 6(x-0)+10(y-2)+2(z-5) & =0 \end{aligned} $
$[\because P(0,2,5)$ line on the plane $]$
$ \Rightarrow \quad 3 x+5 y-10+z-5=0 $
$ \Rightarrow \quad 3 x+5 y+z=15 $
On checking all the options, the option $(4,1,-2)$ satisfy the equation of plane (i).
