Find the distance between the following pairs of points :

(i) (2, 3), (4, 1)

(ii) (-5, 7), (-1, 3)

(iii) (a, b), (-a, -b)

(i) By formula,

Distance between two points (D) = $\sqrt{(y$2 - y1)^2 + (x2 - x1)^2}

Substituting values we get :

$\Rightarrow D = \sqrt{(4 - 2)^2 + (1 - 3)^2} \\[1em] = \sqrt{2^2 + (-2)^2} \\[1em] = \sqrt{4 + 4} \\[1em] = \sqrt{8} \\[1em] = 2\sqrt{2}.$

Hence, distance between (2, 3) and (4, 1) is $2\sqrt{2}$ units.

(ii) By formula,

Distance between two points (D) = $\sqrt{(y$2 - y1)^2 + (x2 - x1)^2}

Substituting values we get :

$\Rightarrow D = \sqrt{(3 - 7)^2 + [-1 - (-5)]^2} \\[1em] = \sqrt{(-4)^2 + [-1 + 5]^2} \\[1em] = \sqrt{16 + 4^2} \\[1em] = \sqrt{16 + 16} \\[1em] = \sqrt{32} \\[1em] = 4\sqrt{2}.$

Hence, distance between (-5, 7) and (-1, 3) is $4\sqrt{2}$ units.

(iii) By formula,

Distance between two points (D) = $\sqrt{(y$2 - y1)^2 + (x2 - x1)^2}

Substituting values we get :

$\Rightarrow D = \sqrt{(-b - b)^2 + (-a - a)^2} \\[1em] = \sqrt{(-2b)^2 + (-2a)^2} \\[1em] = \sqrt{4b^2 + 4a^2} \\[1em] = \sqrt{4(a^2 + b^2)} \\[1em] = 2\sqrt{a^2 + b^2}.$

Hence, distance between (a, b) and (-a, -b) is $2\sqrt{a^2 + b^2}$ units.