Question

If the points $\left( { - 1,3,2} \right)$, $\left( { - 4,2, - 2} \right)$ and $\left( {5,5,\lambda } \right)$ are collinear, then $\lambda $ is equal to
1) $ - 10$
2) 5
3) $ - 5$
4) 10

Answer 1

Hint: Since the points are collinear, then the direction ratios of the points between the line are equal. Also, there will be a constant which will make the direction ratios are equal. Then, find the constant. Next, compare the coordinates and find the value of $\lambda $.

Complete step-by-step answer:
Let the points be $P\left( { - 1,3,2} \right)$, $Q\left( { - 4,2, - 2} \right)$ and $R\left( {5,5,\lambda } \right)$ be collinear points.
Points are collinear, which means that the points are on the straight line.
If the points are on the line, then the direction ratios of the line are equal, which means,
Direction ratio of $PQ$ will be equal to direction ratio of $QR$.
Now, we will calculate the direction ratio of $PQ$
If $\left( {{x_1},{y_1},{z_1}} \right)$ are the coordinates of one point and if $\left( {{x_2},{y_2},{z_2}} \right)$ are the coordinates of other point, then the direction ratio of line is $\left( {{x_2} - {x_1},{y_2} - {y_1},{z_2} - {z_1}} \right)$, then the direction ratio of $PQ$ is,
$\left( { - 4 - \left( { - 1} \right),2 - 3, - 2 - 2} \right) = \left( { - 3, - 1, - 4} \right)$
Now, find the direction ratio of $QR$ is,
$\left( {5 - \left( { - 4} \right),5 - 2,\lambda - \left( { - 2} \right)} \right) = \left( {9,3,\lambda + 2} \right)$
Since, the direction ratios will be equal and let $\alpha $ be the constant between two coordinates.
$\left( { - 3, - 1, - 4} \right) = \alpha \left( {9,3,\lambda + 2} \right)$
On comparing the coordinates, we get,
$
   - 3 = 9\alpha \\
  \alpha = - \dfrac{1}{3} \\
 $
Then, $\left( { - 3, - 1, - 4} \right) = - \dfrac{1}{3}\left( {9,3,\lambda + 2} \right)$
$
   - \dfrac{1}{3}\left( {\lambda + 2} \right) = - 4 \\
   \Rightarrow \lambda + 2 = 12 \\
   \Rightarrow \lambda = 10 \\
 $
Note: The number proportional to the direction cosine is called as the direction ratio of a line. Many students forget to multiply a constant while comparing the coordinates. Also, the sum of the squares of the direction cosines of a line is equal to 1.