Question

How many lines can pass through two given points?

Answer 1

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
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:
$$\begin{array}{rl}& h=\sum _{i=1}^n(n-i)=(n-1)+(n-2)+(n-3)+.....+(n-n)\\ & =\sum _{i=1}^{n}n-\sum _{i=1}^{n}i\Rightarrow L=n^2-{\displaystyle \frac{n(n+1)}{2}}={\displaystyle \frac{2n^2-n^2-n}{2}}\\ & L={\displaystyle \frac{n^2-n}{2}}={\displaystyle \frac{n(n-1)}{2}}\end{array}$$
$$\therefore L={\displaystyle \frac{n(n-1)}{2}}$$, where L = number of lines.
So, for 4 points, n=4,
$$L={\displaystyle \frac{4(4-1)}{2}}={\displaystyle \frac{4\times 3}{2}}=6$$lines.
Where n=1, $$L={\displaystyle \frac{1(1-1)}{2}}={\displaystyle \frac{0}{2}}$$i.e. Infinite number of lines.
Where n=2, $$L={\displaystyle \frac{2(2-1)}{2}}=1$$etc.