Question

If the three points\[(3q,0),\]\[(0,3p)\] and \[(1,1)\]are collinear, then which one is correct?
1) \[(\dfrac{1}{p}) + (\dfrac{1}{q}) = 0\]
2) \[(\dfrac{1}{p}) + (\dfrac{1}{q}) = 1\]
3) \[(\dfrac{1}{p}) + (\dfrac{1}{q}) = 3\]
4) \[(\dfrac{1}{p}) + (\dfrac{3}{q}) = 1\]

Answer 1

Hint: This question can be easily solved by the basic concepts of straight lines like Slope. If the Slope of the lines formed by joining the points is equal then, the points are collinear.
The formula for the Slope of the line is
\[ \Rightarrow m = \dfrac{{({y_2} - {y_1})}}{{({x_2} - {x_1})}}\], where m is the Slope of the line is formed by joining the points \[({x_1},{y_1})\]and \[({x_2},{y_2})\]

Complete step-by-step answer:
We are given three points let’s name them A, B, and C as shown below,
\[ \Rightarrow A = (3q,0)\]
\[ \Rightarrow B = (0,3p)\]
\[ \Rightarrow C = (1,1)\]
Now, let’s find the Slope of the line formed by joining the points A and C,
\[ \Rightarrow {m_1} = \dfrac{{(0 - 1)}}{{(3q - 1)}}\]
\[ \Rightarrow {m_1} = \dfrac{1}{{(1 - 3q)}} - - - - (i)\]
Now, let’s find the Slope of the line formed by joining the points B and C,
\[ \Rightarrow {m_2} = \dfrac{{(3p - 1)}}{{(0 - 1)}}\]
\[ \Rightarrow {m_2} = \dfrac{{(1 - 3p)}}{1} - - - - (ii)\]
After comparing\[(i)\] and \[(ii)\]we get
\[ \Rightarrow \dfrac{1}{{(1 - 3q)}} = \dfrac{{(1 - 3p)}}{1}\]
Now, by cross multiplying, we get,
\[ \Rightarrow 1 = (1 - 3p)(1 - 3q)\]
Opening the R.H.S we get,
\[ \Rightarrow 1 = 1 - 3p - 3q + 9pq\]
Simplifying the equation we get,
\[ \Rightarrow 3p + 3q = 9pq\]
Dividing both the sides by 3,
\[ \Rightarrow p + q = 3pq\]
Now, dividing both the sides of the equation by \[pq\]we get,
\[ \Rightarrow (\dfrac{1}{p}) + (\dfrac{1}{q}) = 3\]
Thus, option(3) is the correct answer.
So, the correct answer is “Option 3”.

Note: This question can have multiple solutions like; if we form a triangle using the three points and try to calculate the area, we would get zero as the answer because collinear points lie on the same line, and from there, we can approach the question as we would have an equation in p and q. So, for such questions, try to form an equation in the variables using some conditions and then simplify the equation to the correct answer. Do not commit calculation mistakes, and be sure of the final answer.