Question

If the points(x,y) is collinear with the points(a,0) and (0,b) prove that x/a + y/b=1

Answer 1

If the three points A,B and C are collinear, then 
slopes of any two pairs of points will be equal.

Now, apply slope formula to find the slopes of the respective pairs of points:

Slope of AB = -y/(a-x)

Slope of BC = b/(-a)

Slope of AC = (b-y)/(-x)

Slope of AB = Slope of AC

-y/(a-x) = (b-y)/(-x)

xy = (a-x) (b-y)

xy = ab -ay -bx + xy

ab = bx + ay

divide by ab on both sides,

1 = x/a + y/b

Answer 2

Let the points are A(x, y), B(a, 0) and C(0, b).

If the points A, B, C are collinear, then the area of the triangle will be equal to zero

=> (1/2)*{x(0 - b) + a(b - y) + 0(y - 0)} = 0

=> -bx + ab - ay = 0

=> bx + ay = ab

=> bx/ab + ay/ab = 1

=> x/a + y/b = 1

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