Question

Find the distance between the following pairs of points:
(i) (2, 3), (4, 1) (ii) (–5, 7), (–1, 3) (iii) (a, b), (–a, –b)

Answer 1

Solutions:-

To Find:-

• Distance between the following points

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

We know the distance formula is as follows:-

• $\dag{\boxed{\underline{\pink{\rm{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}}}}}$

Where:-

• x₁ = abscissa of the first coordinate
• x₂ = abscissa of the second coordinate
• y₁ = ordinate of the first coordinate
• y₂ = ordinate of the second coordinate

Here,

Let the points on both the Coordinates be P and Q, i.e. P(2, 3) and Q(4, 1), Such that:-

$\sf{x_1 = 2\:and\:x_2 = 4}$

$\sf{y_1 = 3\:and\:y_2 = 1}$

Putting all the values in the formula:-

$\sf{PQ = \sqrt{(4-2)^2 + (1-3)^2}}$

$= \sf{PQ = \sqrt{(2)^2 + (-2)^2}}$

$= \sf{PQ = \sqrt{4 + 4}}$

$= \sf{PQ = \sqrt{8}}$

$= \sf{PQ = 2\sqrt{2}}$

∴ The distance between the points P(2, 3) and Q(4, 1) is 2√2 units.

______________________________________

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

⟶ Let A and B be the points on coordinates (-5, 7) and (-1, 3) respectively, such that:-

$\sf{x_1 = -5\:and\:x_2 = -1}$

$\sf{y_1 = 7\:and\:y_2 = 3}$

Putting all the values in the formula:-

$\sf{AB = \sqrt{[(-1) - (-5)]^2 + (3 - 7)^2}}$

$= \sf{AB = \sqrt{(-1 + 5)^2 + (-4)^2}}$

$= \sf{AB = \sqrt{(4)^2 + 16}}$

$= \sf{AB = \sqrt{16 + 16}}$

$= \sf{AB = \sqrt{32}}$

$= \sf{AB = 4\sqrt{2}}$

∴ The distance between the points A(-5, 7) and B(-1, 3) is 4√2 units.

______________________________________

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

⟶ Let C and D be the points on coordinates (a, b) and (-a, -b) respectively, such that:+

$\sf{x_1 = a\:and\:x_2 = -a}$

$\sf{y_1 = b\:and\:y_2 = -b}$

Putting all the values in the formula:-

$\sf{CD = \sqrt{(-a - a)^2 + (-b - b)^2}}$

$= \sf{CD = \sqrt{(-2a)^2 + (-2b)^2}}$

$= \sf{CD = \sqrt{4a^2 + 4b^2}}$

$= \sf{CD = \sqrt{4(a^2 + b^2)}}$

$= \sf{CD = 2\sqrt{a^2 + b^2}}$

∴ The distance between the points C(a, b) and D(-a, -b) is 2√a² + b² units.

______________________________________

Answer 2

Answer:

$\bf\red{Question}$

Find the distance between the following pairs of points:

(i) (2, 3), (4, 1) (ii) (–5, 7), (–1, 3) (iii) (a, b), (–a, –b)

$\bf\red{To\:find}$

We have to find the distance between following pair of points

$\bf\red{Formula}$

$\sqrt{(x_ 2 - x_1) {}^{2} + (y_2 - y_1) } {}^{2}$

$\bf\red{Solution}$

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

$\sqrt{( 2 - 4) {}^{2} + (3 - 1 } {}^{2}$

$\sqrt{(2 - 2) {}^{2} + (2) {}^{2} } = \sqrt{4 + 4 }= \sqrt{8} = 2 \sqrt{2}$

(ii) (–5, 7), (–1, 3)

$\sqrt{( - 5 - ( - 1)) {}^{2} + (7 - 3) {}^{2} }$

$\sqrt{( - 4) {}^{2} + (4) {}^{2} } = \sqrt{16 + 16} = \sqrt{32} = 4 \sqrt{2}$

(iii) (a, b), (–a, –b)

$\sqrt{(a - ( - a)) {}^{2} + (b - ( - b)) {}^{2} }$

$\sqrt{(2a) {}^{2} + (2b) {}^{2} } = \sqrt{4a {}^{2} + 4 {b}^{2} } = 2 \sqrt{a {}^{2} + {b}^{2} }$

_________________________