Question

if that point (a,0) (0,b) (1,1) are collinear ,prove that a+b=ab

Answer 1

Hope this helppp............

Answer 2

The points (a,0) (0,b) (1,1) are collinear then  sum of a and b is equal to the product of a and b i.e a+b = ab.

Step-by-step explanation:

Given: The points (a,0) (0,b) (1,1) are collinear

To find : To prove  a+b = ab

Solution

Here(a,0),(0,b) and (1,1) are collinear

The condition given that three points to be collinear, the area of the triangle formed by these points will be zero.

Vertices of a triangle are $(x_{1} ,y_{1}), (x_{2} ,y_{2})$and ($(x_{3}, y_{3})$ then the area of the triangle is given by

Δ = $\frac{1}{2}\left[\begin{array}{ccc}x_{1} &y_{1} &1 \\x_{2} &y_{2} &1\\x_{3} &y_{3} &1\end{array}\right]$  = 0

On substituting the value from above,

Δ = $\frac{1}{2}\left[\begin{array}{ccc}a &0 &1 \\0 &b &1\\1 &1&1\end{array}\right]$  = 0

On expanding the matrices,

⇒ 0 = $\frac{1}{2} [a\left[\begin{array}{ccc}b&1\\1&1\end{array}\right] - 0 \left[\begin{array}{ccc}0&1\\1&1\end{array}\right] + 1\left[\begin{array}{ccc}0&b\\1&1\end{array}\right]]$

⇒ $\frac{1}{2} [a(b-1)-0(-1)+1(-b)]= 0$

⇒ $\frac{1}{2}[ab - a -b] = 0$

⇒ ab = a+b

⇒ a+b = ab

Final answer:

The point (a,0) (0,b) (1,1) are collinear, then a+b is equal to  ab i.e a+b = ab

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