prove that the points A(a,0),B(0,b)and c(1,1)are collinear if 1/a+1/b=1
Answers
Step-by-step explanation:
Given :-
The points A(a,0),B(0,b)and c(1,1)
To find :-
Prove that the points A(a,0),B(0,b)and c(1,1)are collinear if 1/a+1/b=1
Solution :-
Given points are :A(a,0),B(0,b)and c(1,1)
Let (x1, y1) = (a,0) => x1 = a and y1 = 0
Let (x2, y2) = (0,b) => x2 = 0 and y2 = b
Let (x3, y3) = (1,1) => x3 = 1 and y3 = 1
If the given points are Collinear points then the area of the triangle formed by the points is Zero.
We know that
Area of a triangle formed by the points (x1, y1), (x2, y2) and (x3, y3) is ∆ =
(1/2) | x1(y2-y3)+x2(y3-y1)+x3(y1-y2) | sq.units
We have ∆ = 0
On Substituting these values in the above formula then
=> ∆ = (1/2) | a(b-1)+0(1-0)+1(0-b) | = 0
=> (1/2) | a(b-1)+0(1)+1(-b) | = 0
=> (1/2) | a(b-1) +0+(-b) | = 0
=> (1/2) | a(b-1)-b | = 0
=> (1/2) | ab-a-b | = 0
=> (1/2)(ab-a-b) = 0
=> ab-a-b = 0×2
=> ab-a-b = 0
=> ab -(a+b) = 0
=> ab = a+b
=> a+b = ab
On dividing by 'ab' both sides then
=> (a+b)/ab = (ab/ab)
=> (a+b)/ab = 1
=> (a/ab)+(b/ab) = 1
=> (1/b) + (1/a) = 1
=> 1/a + 1/b = 1
Hence, Proved.
Answer :-
If the points A(a,0),B(0,b)and c(1,1) are collinear then 1/a+1/b=1.
Used Concept:-
If the given points are Collinear points then the area of the triangle formed by the points is Zero.
Used formulae:-
Area of a triangle formed by the points (x1, y1), (x2, y2) and (x3, y3) is ∆ =
(1/2) | x1(y2-y3)+x2(y3-y1)+x3(y1-y2) | sq.units