Question

If the point $A(9,8, - 10),B(3,2, - 4)$ and $C(5,4, - 6)$ be collinear, Then the point C divides The line AB in the ratio
A. $2:1$
B. $3:1$
C. $1:2$
D. $ - 1:2$

Answer 1

Hint: Given, the point $A(9,8, - 10),B(3,2, - 4)$ and $C(5,4, - 6)$ be collinear. We have to find the ratio in which C divides the line AB. We will assume the ratio $k:1$. We apply section formula $(x,y,z) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$ then compare points with C to get the value of k.

Formula used: Let a point P divides the line PQ in the ratio $m:n$, then
Section formula$(x,y,z) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$

Complete step by step solution:
Given, the point $A(9,8, - 10),B(3,2, - 4)$ and $C(5,4, - 6)$ be collinear.
Let us assume point C divides the line AB in the ratio $k:1$
To find the value of k we will use the section formula.
Let a point P divides the line PQ in the ratio $m:n$, then
Section formula$(x,y,z) = \left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}},\dfrac{{m{z_2} + n{z_1}}}{{m + n}}} \right)$
Comparing with the above formula
$(x,y,z) = (5,4, - 6)$
$({x_1},{y_1},{z_1}) = (9,8, - 10)$
$({x_2},{y_2},{z_2}) = (3,2, - 4)$
$m = k,\,n = 1$
Putting the above values in the section formula
$(5,4, - 6) = \left( {\dfrac{{3k + 9}}{{k + 1}},\dfrac{{2k + 8}}{{k + 1}},\dfrac{{ - 4k - 10}}{{k + 1}}} \right)$
After comparing
$5 = \dfrac{{3k + 9}}{{k + 1}}$
Multiplying both sides with $k + 1$
$5(k + 1) = 3k + 9$
After simplification
$5k + 5 = 3k + 9$
Shifting terms with k on one side and terms without k on another side
$5k - 3k = 9 - 5$
After solving the above equation
$2k = 4$
Dividing both sides with $2$
$k = 2$
Hence, point C divides the line AB in $2:1$

So, option (A) is the correct answer.

Note: Students should assume a simple ratio to get a smooth calculation. Apply the section formula correctly in order to get error-free calculation. After applying, the section formula should compare the point with C not with the other because C divides AB and does the calculation correctly to get the required ratio.