Question

The coordinate of point in which the line joining the point $(2,5, - 7)$ and $( - 3, - 1,8)$ are intersected by the y-z planes are
A. $(0,\dfrac{{13}}{5}, - 1)$
B. $(0,\dfrac{{ - 13}}{5}, - 2)$
C. $(0,\dfrac{{ - 13}}{5},\dfrac{2}{5})$
D. $(0,\dfrac{{13}}{5},\dfrac{2}{5})$

Answer 1

Hint: Given, the line joining the point $(2,5, - 7)$ and $( - 3, - 1,8)$. We have to find the coordinate of the point which is the intersection point of the line joining the point $(2,5, - 7)$ and $( - 3, - 1,8)$ and the y-z plane. First, we will find the equation of the line and then use the concept that when the line intersects the y-z plane x-coordinate is zero to find the coordinates points.

Formula used:
Equation of line joining two points $\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}$

Complete step by step solution:
Given, the line joining the point $(2,5, - 7)$ and $( - 3, - 1,8)$.
Equation of line joining two points $\dfrac{{x - {x_1}}}{{{x_2} - {x_1}}} = \dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{z - {z_1}}}{{{z_2} - {z_1}}}$
So, using the above formula
$\dfrac{{x - 2}}{-5} = \dfrac{{y - 5}}{-6} = \dfrac{{z + 7}}{{15}}$
Multiplying by -1,
$\dfrac{{x - 2}}{5} = \dfrac{{y - 5}}{6} = \dfrac{{z + 7}}{{ - 15}}$
When the line meets y-z plane
$ \Rightarrow x = 0$
$ \Rightarrow \dfrac{{ - 2}}{5} = \dfrac{{y - 5}}{6} = \dfrac{{z + 7}}{{ - 15}}$
Comparing first two
$\dfrac{{ - 2}}{5} = \dfrac{{y - 5}}{6}$
Cross multiplying
$ - 2 \times 6 = 5(y - 5)$
After solving
$ - 12 = 5y - 25$
Shifting variable on one side and constant on other
$5y = - 12 + 25$
$5y = 13$
Dividing both sides by 5
$y = \dfrac{{13}}{5}$
Comparing last two
$\dfrac{{ - 2}}{5} = \dfrac{{z + 7}}{{ - 15}}$
Cross multiplying
$30 = 5z + 35$
Shifting variable on one side and constant on other
$5z = - 35 + 30$
$5z = - 5$
Dividing both sides by 5
$z = - 1$
Hence, $(x,y,z) = (0,\dfrac{{13}}{5}, - 1)$

So, option (A) is the correct answer.

Note: Students should find the equation of the line correctly so that they can do further calculations correctly. And should understand the concept that when a line intersects the y-z plane means x-coordinate is zero. This will help to find the value of y and z. They should pay attention to every detail while solving question.