How many lines can pass through two distinct points?

Question

Vedantu Content Team · Accepted Answer

Hint: Slope of a line passing through two distinct points $\left( {{x}_{1}},{{y}_{1}} \right)\ and\ \left( {{x}_{2}},{{y}_{2}} \right)$ will be unique and be equal to $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. And a line passing through a point with a given slope is unique. So, only one line can pass through two distinct 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 two lines AB and CD pass 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 given points.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.

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