Question

If (a,0) ,(0,b) and (1,1) are collinear than prove that a+b = ab

Answer 1

Answer:

Here given that,

x1 = a

x2 = 0

x3 = 1

y1 = 0 , y2 = b, y3 = 1 Hence it is in collinear that's why we taken as 0

a (b-1) - 0 + 1(-b) = 0

ab - a - b = 0

ab = a + b

Answer 2

SOLUTION:-

Let A(a,0), B(0,b) & C(1,1) be the given points.

Suppose all given points are collinear.

∴Area of ∆ABC = 0

$= > \frac{1}{2} |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| = 0 \\ \\ = > \frac{1}{2} |a(b - 1) + 0(1 - 0) + 1(0 - b)| = 0 \\ \\ = > \frac{1}{2} |ab - a - b| = 0 \\ \\ = > ab - a - b = 0$

Dividing both sides by ab, we get,

$= > \frac{ab}{ab} - \frac{a}{ab} - \frac{b}{ab} = 0 \\ \\ = > 1 - \frac{1}{b} - \frac{1}{a} = 0 \\ \\ = > \frac{1}{a} + \frac{1}{b} = 1$

Hence, the given points are collinear only if when,

$= > \frac{1}{a} + \frac{1}{b} = 1.$

Hence, proved

Hope it helps ☺️