Question

Find the ratio in which the plane \[x - 2y + 3z = 17\] divides the line joining the points \[\left( { - 2,4,7} \right)\] and \[\left( {3, - 5,8} \right)\].
A. \[10:3\]
B. \[3:1\]
C. \[3:10\]
D. \[10:1\]

Answer 1

Hint: Here, an equation of the plane and line joining the two points are given. Use the formula of the ratio in which the plane divides the line segment. Substitute the values and solve it to get the required ratio.

Formula used: The formula of the ratio in which the plane \[ax + by + cz + d = 0\] divides the line joining the points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\] is: \[\dfrac{{ - \left( {a{x_1} + b{y_1} + c{z_1} + d} \right)}}{{a{x_2} + b{y_2} + c{z_2} + d}}\]

Complete step by step solution: Given: The plane \[x - 2y + 3z = 17\] divides the line joining the points \[\left( { - 2,4,7} \right)\] and \[\left( {3, - 5,8} \right)\].

We have to calculate the ratio.
So, we use the formula of the ratio in which the plane \[ax + by + cz + d = 0\] divides the line joining the points \[\left( {{x_1},{y_1},{z_1}} \right)\] and \[\left( {{x_2},{y_2},{z_2}} \right)\].
Comparing the given information with above formula, we get
\[a = 1,b = - 2,c = 3,d = - 17\] , \[{x_1} = - 2,{y_1} = 4,{z_1} = 7\], and \[{x_2} = 3,{y_2} = - 5,{z_2} = 8\]

Now substitute the values in the formula \[\dfrac{{ - \left( {a{x_1} + b{y_1} + c{z_1} + d} \right)}}{{a{x_2} + b{y_2} + c{z_2} + d}}\].
The required ratio is:
\[R = \dfrac{{ - \left( {\left( 1 \right)\left( { - 2} \right) + \left( { - 2} \right)\left( 4 \right) + \left( 3 \right)\left( 7 \right) - 17} \right)}}{{\left( 1 \right)\left( 3 \right) + \left( { - 2} \right)\left( { - 5} \right) + \left( 3 \right)\left( 8 \right) - 17}}\]
\[ \Rightarrow R = \dfrac{{ - \left( { - 2 - 8 + 21 - 17} \right)}}{{3 + 10 + 24 - 17}}\]
\[ \Rightarrow R = \dfrac{{ - \left( { - 6} \right)}}{{20}}\]
\[ \Rightarrow R = \dfrac{6}{{20}}\]
\[ \Rightarrow R = \dfrac{3}{{10}}\]
\[ \Rightarrow R = 3:10\]
Thus, the ratio in which the plane \[x - 2y + 3z = 17\] divides the line joining the points \[\left( { - 2,4,7} \right)\] and \[\left( {3, - 5,8} \right)\] is \[3:10\].

Thus, Option (C) is correct.

Note: We can also solve this question by using the section formula \[\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)\] .
Substitute coordinates of the points, \[m = k\] and \[n = 1\] in the formula and solve it. Then substitute the values in the equation of the plane and solve the equation. The value of the variable \[k\] is the required ratio.