Question

Number of lines that can be drawn from distinct points on a plane is:
A. Two
B. One
C. Zero
D. Infinite

Answer 1

Hint: It is an axiom that only one line can pass through two given points. We can prove this by using contradiction. Assume two lines passing through two points on a plane. Then prove that our assumption is wrong and two different lines can’t pass through two given points.

Complete step-by-step answer:
Let us assume two points $\left( {{x}_{1}},{{y}_{1}} \right)\ and\ \left( {{x}_{2}},{{y}_{2}} \right)$ and let two lines AB and CD pass through these two points.
We know that slope of a line passing through two points $\left( {{x}_{1}},{{y}_{1}} \right)\ and\ \left( {{x}_{2}},{{y}_{2}} \right)$ is given by,
$m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Thus, slope of $AB=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$ and slope of $CD=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
Both of these lines are passing through $\left( {{x}_{1}},{{y}_{1}} \right)\ and\ \left( {{x}_{2}},{{y}_{2}} \right)$ .
We can see that, (slope of AB) = (slope of CD)
We know that two different lines passing through a given point cannot have the same slope.
So, the lines AB and CD cannot be different.
Thus, our assumption is wrong and AB = CD.
Hence, one and only one unique line can pass through two fixed points.
Therefore, the number of lines that can be drawn from two distinct points is one and option (C) is the correct answer.

Note: In the solution, we have mentioned that two different lines passing through a fixed point cannot have the same slopes. Be careful that two different lines can have the same slope but if two different lines are passing through a fixed point, they will have different slopes.