Q) If points (a, 0), (0, b), (1,1) are collinear, then 1/a + 1/b is ?

Ans:  [I have solved the question by two methods. Please choose the method whichever is in line with your syllabus.]

Method 1:

For points to be collinear, area of triangle formed by the three points is equal to 0.

We know that the area of a triangle formed by the points (x₁, y₁), (x₂, y₂), (x₃, y₃) is given by:

Hence, the area of the triangle formed by the points (a, 0), (0, b), (1,1) will be:

[a (b – 1) + 0 (1 – 0) + 1 (0 – b)] = 0

∴ [a (b – 1) – b] = 0

∴ a b – a – b = 0

∴ a + b = ab

By dividing ab on both sides, we get:

∴

∴

Method 2:

Let’s consider that given three points [A (a, 0), B (0, b), C (1,1)] are collinear and point B divides the line AC in ratio of k:1

We know that, by section formula, if a point (x, y) divides the line joining the points (x₁, y₁​) and (x₂​, y₂​) in the ratio m : n, then the coordinates of intersection point (x, y) is given by:

,

Therefore, coordinates of points B, dividing line AC in ratio of k : 1 will be:

B(x,y) =

B(x,y) =

Since coordinates of B are given as (0, b)

By comparing x coordinate (or abscissa), we get:

= 0

∴ k + a = 0

∴ k = – a

By comparing y coordinate (or ordinate), we get:

= b

∴ k = b (k + 1)

since k = – a

∴ – a = b (- a + 1)

∴ – a = – a b + b

∴ a + b = ab

By dividing ab on both sides, we get:

∴

∴

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