Prove that the points a , 0 , 0 , b and 1 , 1 are collinear, if 1 a + 1 b = 1 .

Let, A x 1 , y 1 = a , 0 , B x 2 , y 2 = 0 , b and C x 3 , y 3 = 1 , 1 We have, 1 a + 1 b = 1 ⇒ b + a a b = 1 ⇒ b + a = a b ⇒ a b - a - b = 0 ....... 1 Now, Area of △ A B C = 1 2 x 1 y 2 - y 3 + x 2 y 3 - y 1 + x 3 y 1 - y 2 = 1 2 a b - 1 + 0 1 - 0 + 1 0 - b = 1 2 a b - a + 0 - b = 1 2 a b - a - b = 1 2 × 0 [From 1 = 0 Hence, the points A , B , C i.e. the given points, are collinear.

If the points x 1 , y 1 , x 2 , y 2 , x 3 , y 3 lie on the same line, prove that y 2 - y 3 x 2 x 3 + y 3 - y 1 x 3 x 1 + y 1 - y 2 x 1 x 2 = 0 .

Let, the coordinates of A , B , C be x 1 , y 1 , x 2 , y 2 , x 3 , y 3 . Since, A , B and C lie on the same line, so they are collinear. ∴ Area of △ A B C = 0 ⇒ 1 2 x 1 y 2 - y 3 + x 2 y 3 - y 1 + x 3 y 1 - y 2 = 0 ⇒ x 1 y 2 - y 3 + x 2 y 3 - y 1 + x 3 y 1 - y 2 = 0 ⇒ x 1 y 2 - y 3 + x 2 y 3 - y 1 + x 3 y 1 - y 2 x 1 x 2 x 3 = 0 [dividing both sides by x 1 x 2 x 3 ] ⇒ x 1 y 2 - y 3 x 1 x 2 x 3 + x 2 y 3 - y 1 x 1 x 2 x 3 + x 3 y 1 - y 2 x 1 x 2 x 3 = 0 ⇒ y 2 - y 3 x 2 x 3 + y 3 - y 1 x 3 x 1 + y 1 - y 2 x 1 x 2 = 0 Hence, proved.

E M B I B E

Mathematics for Class 10>Chapter 11 - Coordinate Geometry>EXERCISE 11.2

Manipur Board Solutions for Chapter: Coordinate Geometry, Exercise 2: EXERCISE 11.2

Author:Manipur Board

Manipur Board Mathematics Solutions for Exercise - Manipur Board Solutions for Chapter: Coordinate Geometry, Exercise 2: EXERCISE 11.2

Attempt the practice questions on Chapter 11: Coordinate Geometry, Exercise 2: EXERCISE 11.2 with hints and solutions to strengthen your understanding. Mathematics for Class 10 solutions are prepared by Experienced Embibe Experts.

Questions from Manipur Board Solutions for Chapter: Coordinate Geometry, Exercise 2: EXERCISE 11.2 with Hints & Solutions

EASY

10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 11 - Coordinate Geometry>EXERCISE 11.2>Q 4

Find the value of $x$ for which the following points are collinear.

$(\frac{- 1}{2}, 3), (- 5, 6), (- 8, x)$

EASY

10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 11 - Coordinate Geometry>EXERCISE 11.2>Q 4

Find the value of $x$ for which the following points are collinear.

$(x, 5), (2, - 5), (4, - 9)$

EASY

10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 11 - Coordinate Geometry>EXERCISE 11.2>Q 5

If the points $(x, y), (- 5, 7)$ and $(- 4, 5)$ are collinear, then show that $2 x + y + 3 = 0$ .

MEDIUM

10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 11 - Coordinate Geometry>EXERCISE 11.2>Q 6

Prove that the points $(a, 0), (0, b)$ and $(1, 1)$ are collinear, if $\frac{1}{a} + \frac{1}{b} = 1$ .

MEDIUM

10th Manipur Board

IMPORTANT

Mathematics for Class 10>Chapter 11 - Coordinate Geometry>EXERCISE 11.2>Q 7

If the points $(x_{1}, y_{1}), (x_{2}, y_{2}), (x_{3}, y_{3})$ lie on the same line, prove that $\frac{y_{2} - y_{3}}{x_{2} x_{3}} + \frac{y_{3} - y_{1}}{x_{3} x_{1}} + \frac{y_{1} - y_{2}}{x_{1} x_{2}} = 0$ .