Question

Find the ratio in which the YZ-plane divides the line segment formed by joining the points $( - 2,4,7)$ and $(3, - 5,8)$.

Answer 1

Here we are asked to find the ratio in which the YZ-plane divides the line formed by joining the given points. To find the ratio we will use the section formula by substituting the given data in that. Then we are given that the plane is the YZ-plane so the x coordinate will be zero using this we will find the ratio.
Formula: Formula that we need to know:
The point (coordinates) that divides the line joining $\left( {{x_1},{y_1},{z_1}} \right)\& \left( {{x_2},{y_2},{z_2}} \right)$ in the ratio $k:1$ $\left( {\dfrac{{k{x_2} + {x_1}}}{{k + 1}},\dfrac{{k{y_2} + {y_1}}}{{k + 1}},\dfrac{{k{z_2} + {z_1}}}{{k + 1}}} \right)$

Complete step by step answer:
It is given that the line joining by the points $( - 2,4,7)$ and $(3, - 5,8)$ is in YZ-plane and it is being cut in some ratio. We aim to find the ratio that the line has been divided in the YZ-plane.
We know that the point (coordinates) that divides the line joining $\left( {{x_1},{y_1},{z_1}} \right)\& \left( {{x_2},{y_2},{z_2}} \right)$ in the ratio $k:1$ is $\left( {\dfrac{{k{x_2} + {x_1}}}{{k + 1}},\dfrac{{k{y_2} + {y_1}}}{{k + 1}},\dfrac{{k{z_2} + {z_1}}}{{k + 1}}} \right)$
Let us substitute the points that have been given in the question.
We have $({x_1},{y_1},{z_1}) = ( - 2,4,7)$ and $({x_2},{y_2},{z_2}) = (3, - 5,8)$substituting these in the section formula we get
$\left( {\dfrac{{k\left( 3 \right) + \left( { - 2} \right)}}{{k + 1}},\dfrac{{k\left( { - 5} \right) + 4}}{{k + 1}},\dfrac{{k\left( 8 \right) + 7}}{{k + 1}}} \right)$
On simplifying the above, we get
$\left( {\dfrac{{3k - 2}}{{k + 1}},\dfrac{{ - 5k + 4}}{{k + 1}},\dfrac{{8k + 7}}{{k + 1}}} \right)$
It is given that the line is in the YZ-plane so the x coordinate will be equal to zero.
That implies $\dfrac{{3k - 2}}{{k + 1}} = 0$
Let us simplify the above, to find the ratio.
$\Rightarrow 3k - 2 = 0$
$\Rightarrow 3k = 2$
$\Rightarrow k = \dfrac{2}{3}$
Since k:1 is the ratio, it can be written in the form $2:3$. Thus, we have found the ratio in which the line is divided in the YZ-plane.

Note:
If the given line passes through the origin in that case, we can find the ratio by using the line equation also. The ratio is nothing but the relationship between two same kinds of quantities in the division and it can be written as $\dfrac{a}{b} \Rightarrow a:b$. We must give more attention while substituting the values in given points that are the line joining points.