Question

if the points (a, 0), (0, b) and (1, 1) a r e collinear, then prove that 1/a + 1/b=1

Answer 1

Answer:

Explanation:

Given :

• The points (a, 0) , (0 , b) & (1, 1) are collinear.

To Prove :

• 1/a + 1/b = 1

Proof :

Given that, The points (a, 0) , (0 , b) & (1, 1) are collinear.

.°. Area of triangle formed by these points is equal to 0.

=> ¹/2 [x¹(y² - y³) + x²(y³ - y¹) + x³(y¹ - y²)] = 0

Given that, Points are (a, 0) , (0 , b) & (1, 1).

Where,

• (x¹ , y¹) = (a, 0)
• (x² , y²) = (0 , b)
• (x³ , y³) = (1 , 1)

Substitute all values in above formula,

=> ¹/2 [a(b - 1) + 0(1 - 0) + 1(0 - b)] = 0

=> ¹/2 [ab - a - b] = 0

=> ab - a - b = 0 × 1/2

=> ab - a - b = 0

=> -(a + b) = -ab

=> a + b = ab

Divide by ab on both sides,

=> a/ab + b/ab = ab/ab

=> 1/b + 1/a = 1

=> 1/a + 1/b = 1

• Hence Proved!!