Question

Find the distance between the following pair of points:
$(a,0)$ and $(0,b)$

Answer 1

Hint: We have to find the distance between the given pair of points, for that we will apply the distance formula which is equal to the square root of the sum of the squares of the difference in the coordinates of the two points and then we will replace the coordinates given in the formula with the coordinates given in the question. After putting the values of the coordinates of the two points in the formula, we will get the required distance between the given pair of points.

Complete step-by-step answer:
The given points are:-
(a,0) and (0,b)
We know the distance of the point $({{x}_{1}},{{y}_{1}})$from $({{x}_{2}},{{y}_{2}})$ $=\sqrt{\left[ {{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}} \right]}$
We have to find the distance between the points $(a,0)\And (0,b)$. For that, we will replace $x_1$ with a, $y_1$ with 0, $x_2$ with 0 and $y_2$ with b.
So we will put the replaced value in the distance formula.
Distance of the point $(a,0)$ from $(0,b)$ $=\sqrt{\left[ {{\left( 0-a \right)}^{2}}+{{\left( b-0 \right)}^{2}} \right]}$
Applying exponents on the bases, we get
Distance of the point $(a,0)$from $(0,b)$ $=\sqrt{0+{{a}^{2}}-2\times 0\times a+{{b}^{2}}+0-2\times 0\times b}$
On further simplification of the equation, we get
Distance of the point $(a,0)$from $(0,b)$ $=\sqrt{{{a}^{2}}+{{b}^{2}}}$
Further simplification is not possible as there are no like terms inside the root.
So the distance between the points $(a,0)\And (0,b)$is $\sqrt{{{a}^{2}}+{{b}^{2}}}$

Note: The formula which we have used here for finding the distance between the two points is in two dimensional.
The distance formula for three dimensional is different.
Say we have to find the distance between the points $({{x}_{1}},{{y}_{1}},{{z}_{1}})$ and $({{x}_{2}},{{y}_{2}},{{z}_{2}})$
So the distance between these two points is
$\sqrt{\left[ {{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}} \right]}$
The only difference in the two dimensional and three dimensional formula of finding distance between the two points is the addition of the square of the difference of the z coordinates.