Question

The value of p for which the points $( - 1,\ 3)$, $(2,\ p)$ and $(5,\ 1)$ are collinear then $p$ is
A. $-1$
B. $2$
C. $1$
D. $-2$

Answer 1

In this question, we need to find the value of $p$ in the points $( - 1,\ 3)$, $(2,\ p)$ and $(5,\ 1)$. Also given that the given points are collinear. First, we need to know the concept of the collinearity of the points. That is, if the given points are said to be collinear, then the slope of the line joining the two points is equal to the slope of the line joining the other points. By using this fact we can easily find the value of $p$.

Complete step by step answer:
Given $( - 1,\ 3)$, $(2,\ p)$ and $(5,\ 1)$. Let us consider the points as $A( - 1,\ 3)$ , $B(2,\ p)$ and $C(5,\ 1)$. Here we need to find the value of p. Given that the three points are collinear it means that the points are on the same line. If the given points are said to be collinear, then the slope of the line joining the points A and B is equal to the slope of the line joining the points A and C .

Now , If the points A, B and C are collinear, then the slope of the line joining the points $( - 1,\ 3)$ and $(2,\ p)$ is equal to the slope of the line joining the points $( - 1,\ 3)$ and $(5,\ - \ 1)$.
Slope of two points is $\dfrac{y_{2} – y_{1}}{x_{2} – x_{1}}$
Since the points are collinear, slope AB will be equal to the slope of AC.
$\Rightarrow \dfrac{p – 3}{2 – ( - 1)} = \dfrac{- 1 – 3}{5 + 1}$
On simplifying we get,
$\Rightarrow \dfrac{p – 3}{3} = \dfrac{- 4}{6}$

On cross multiplying,
We get,
$\Rightarrow \ 6(p – 3)\ = 3( - 4)$
Now on simplifying,
We get,
$6p – 18 = - 12$
On adding both sides by $18$ ,
We get,
$6p = - 12 + 18$

On simplifying,
We get,
$6p = 6$
On dividing both sides by $6$ ,
We get,
$\Rightarrow \ p = \dfrac{6}{6}$
On simplifying,
We get,
$\therefore \ p = 1$
Thus we get the value of $p$ as $1$ .

Hence, option C is the correct answer.

Note: We can also solve these types of questions using slope, geometry and vector methods. We have used slope method here by using the fact that the slope of the line joining the points A and B is equal to the slope of the line joining the points A and C . In the geometric method, we need to find the straight line equation joining the points A and C and substitute point B in it. In the vector method, we use the fact that the vectors AB and AC are parallel to each other and differ only by a constant multiple to solve for $p$.