
How many lines can pass through two distinct points?
Answer
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Hint: Slope of a line passing through two distinct points will be unique and be equal to . 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 and two lines AB and CD pass through two points is given by,
Thus, slope of and slope of .
Both of these lines are passing through .
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.
Complete step-by-step answer:
Let us assume two points
Thus, slope of
Both of these lines are passing through
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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