# How many lines can pass through two given points?

Last updated date: 21st Mar 2023

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**Hint:**Infinite number of lines can pass through a single point. Similarly, an infinite number of curves can pass through 2 points, which are not straight lines. Only 1 straight line can pass through 2 points.

**Complete step-by-step answer:**

Let us consider the 2 points as A and B.

Now, infinite lines can pass through the point A as shown below.

Similarly, infinite lines can pass through point B.

Now, Let's take two points A and B together will have one line passing through it.

Only through Point A \[\to \] Infinite lines.

Only through Point B \[\to \] Infinite lines.

Through A and B \[\to \] 1

Through A and B \[\to \] 1

**Number of straight lines that can pass through 2 points A and B = 1.**

**Note:**

Let us consider a case of non-collinear points. If we have been given 4 points and need to find how many lines can pass through these points.

The method is to find the number of straight lines that can be formed.

3 Lines 5 Lines 6 Lines

\[\therefore \](4 - 1) (4 - 1) + (4 - 2) (4 - 1) + (4 - 2) + (4 - 3) + (4 - 4)

\[\therefore \]They are of the form:

\[\begin{align}

& h=\sum\limits_{i=1}^{n}{\left( n-i \right)}=\left( n-1 \right)+\left( n-2 \right)+\left( n-3 \right)+.....+\left( n-n \right) \\

& =\sum\limits_{i=1}^{n}{n}-\sum\limits_{i=1}^{n}{i}\Rightarrow L={{n}^{2}}-\dfrac{n\left( n+1 \right)}{2}=\dfrac{2{{n}^{2}}-{{n}^{2}}-n}{2} \\

& L=\dfrac{{{n}^{2}}-n}{2}=\dfrac{n\left( n-1 \right)}{2} \\

\end{align}\]

\[\therefore L=\dfrac{n\left( n-1 \right)}{2}\], where L = number of lines.

So, for 4 points, n=4,

\[L=\dfrac{4\left( 4-1 \right)}{2}=\dfrac{4\times 3}{2}=6\]lines.

Where n=1, \[L=\dfrac{1\left( 1-1 \right)}{2}=\dfrac{0}{2}\]i.e. Infinite number of lines.

Where n=2, \[L=\dfrac{2\left( 2-1 \right)}{2}=1\]etc.

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