Question

find the distance between the points (a,2) and (3,4)​

Answer 1

Given:

Two points are given:

• (a,2)
• (3,4)

To Find:

• Distance between the given points.

Assumption:

Let,

• A = (a,2)
• B = (3,4)

Formula used:

To find the distance AB, we will be using the distance formula:

$\bigstar\boxed{\tt{AB = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}}$

Solution:

From the given points, we know that:

• $\sf{x_1 = a}$

• $\sf{x_2 = 3}$

• $\sf{y_1 = 2}$

• $\sf{y_2 = 4}$

By applying our values in the distance formula, we get:

$\tt{\longmapsto AB = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}}$

$\tt{\longmapsto AB = \sqrt{(3-a)^2+(4-2)^2}}$

$\tt{\longmapsto AB = \sqrt{(3-a)^2+(2)^2}}$

$\tt{\longmapsto AB = \sqrt{(3-a)^2+4}}$

We know that,

• $\bf{(a-b)^2 = a^2+b^2-2ab}$

We are going to use this identity in (3-a)²:

$\tt{\longmapsto AB = \sqrt{(3^2+a^2-2 \times 3 \times a)+4}}$

$\tt{\longmapsto AB = \sqrt{(9+a^2-6a)+4}}$

$\tt{\longmapsto AB = \sqrt{13+a^2-6a}}$

$\tt{\longmapsto AB = \sqrt{a^2-6a+13}}$

Conclusion:

Hence, the distance between the points (a,2) and (3,4) is:

$\longmapsto{\boxed{\tt{\sqrt{a^2-6a+13}}}}$

Note:

• Further, we can't factorize the acquired equation since their roots are unreal.
• So, we have stopped our process of solving and it would be our required distance between the points.