Question

Find the ratio in which the $YZ$ plane divides the line joining A (2,4,5) and B (3,5, -4). Also find the points of intersection.

Answer 1

Hint:In this question we know that there is a point which divides a line into 2 unequal parts. We can assume its coordinates and apply the section formula. The ratio in which the point divides the given line segment can be found if we know the coordinates of that point.

Formula used:
\[x = \dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}} \\
\Rightarrow y = \dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}} \\
\Rightarrow z = \dfrac{{{m_1}{z_2} + {m_2}{z_1}}}{{{m_1} + {m_2}}} \]

Complete step by step answer:
According to the question there is a line AB with coordinates A (2,4,5) and B (3,5, -4) which is divided unequally into 2 parts, let the point of the plane that divides the line be P with coordinates (0, Y,Z) since the plane is Y-Z plane hence X coordinate will be 0. And the ratio in which it line segment AB will be divided be \[\lambda :1\] somewhat like this

The point P divides the segment into \[\lambda :1\]. Section formula is being used when A and B be the given two points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] respectively, and P be the point dividing the line-segment AB internally in the ratio \[{m_1}:{m_2}\] then form the sectional formula for determining the coordinate of a point P is given by:
\[x = \dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}},y = \dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}},z = \dfrac{{{m_1}{z_2} + {m_2}{z_1}}}{{{m_1} + {m_2}}}\]
Here \[{m_1}:{m_2}\]=\[\lambda :1\]

Now putting the values
\[ \Rightarrow 0 = \dfrac{{3\lambda + 2}}{{\lambda + 1}}\](since the value of X is 0) - equation 1
\[\Rightarrow y = \dfrac{{5\lambda + 4}}{{\lambda + 1}}\,\,\,\,\,\, - eq\,2 \\
\Rightarrow z = \dfrac{{4\lambda + 5}}{{\lambda + 1}}\,\,\,\,\,\, - eq\,3 \\ \]
Solving equation 1 we get
\[\Rightarrow 3\lambda + 2 = 0 \\
\Rightarrow 3\lambda = - 2 \\
\therefore \lambda = \dfrac{{ - 2}}{3} \\ \]
Therefore, the ratio comes out to be \[ - 2:3\]. The point of intersection P can be calculated by putting the value of lambda in equation 2 and 3, all the coordinates of point P can be found. By solving the values come out to be (0,2,23).

Hence, the point of intersection is (0,2,23).

Note: There are two types of intersection that is internal section formula (if the point of intersection lies in between points A and B, as in this case), here above used section formula is used, there is another section formula that is external section formula that is \[x = \dfrac{{{m_1}{x_2} - {m_2}{x_1}}}{{{m_1} - {m_2}}},y = \dfrac{{{m_1}{y_2} - {m_2}{y_1}}}{{{m_1} - {m_2}}},z = \dfrac{{{m_1}{z_2} - {m_2}{z_1}}}{{{m_1} - {m_2}}}\] .