Question

Find out the ratio in which XOZ plane divides the joining of the points \[\left( 2,3,1 \right)\] and $\left( 6,7,1 \right)$.
A. 3:7 B. 2:7 C. -3:7 D. -2:7

Answer 1

Hint:We try to draw the figure from the given conditions. We try to assume the ratio. Then we try to find out the coordinates of the intersecting point of the plane with the line segment. The y-coordinate of the point will be 0 as the point is on the XOZ plane. We equate both sides to get a linear equation of p. we solve that to find the ratio.

Complete step by step answer:
Let’s assume the plane XOZ divides the joining line of the points \[\left( 2,3,1 \right)\] and $\left( 6,7,1 \right)$ in the ratio p:1.
So, the intersecting point of the plane and the joining line will be on the plane.

Let’s consider the figure where joining line of $A\left( 2,3,1 \right)$ and $B\left( 6,7,1 \right)$ intersect the plane XOZ at M.
Now, we know that all the points on the plane XOZ has one thing constant.
The plane call be called as XZ plane also. So, the y-coordinate will always be 0.
Now, we consider the ratio of p:1 in which the point M divides the line AB internally.
We know that when a point with coordinates $\left( f,g,h \right)$ divides the points $\left( {{x}_{1}},{{y}_{1}},{{z}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}},{{z}_{2}} \right)$ in the ratio of m:n internally then we can say
$f=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},g=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n},h=\dfrac{m{{z}_{2}}+n{{z}_{1}}}{m+n}$.
We apply this theorem on the given figure of line AB to find its y-coordinate.
The y-coordinate of point M will be $\dfrac{p\times \left( 7 \right)+1\times \left( 3 \right)}{p+1}$ .
This coordinate value will be equal to 0.
So, $\dfrac{p\times \left( 7 \right)+1\times \left( 3 \right)}{p+1}=\dfrac{7p+3}{p+1}=0$
We solve the equation to get value of p.
\[\begin{align}
  & \dfrac{7p+3}{p+1}=0 \\
 & \Rightarrow 7p+3=0 \\
 & \Rightarrow p=\dfrac{-3}{7} \\
\end{align}\]
So, the ration is $\dfrac{-3}{7}:1=-3:7$. The correct option is (C).

Note:
We need to remember the negative part appears as the point intersects the line externally. From the start we assumed that it intersected internally. So, its sign changed to negative. We need not to worry about the sign as it’s self-explanatory. We need to remember that the intersection of a plane and a line will always will be a point not a line.