Question

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Answer 1

Answer:

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Answer 2

The points (a,a²), (b,b²) and (c,c²)  can never be collinear

Step-by-step explanation:

Given data

$(x_1, y_1) = (a, a^{2})$

$(x_2, y_2) = (b, b^{2} )$

$(x_3, y_3) = (c, c^{2})$

To prove - that given points can never be collinear

We have to find the distance between the points to find whether they are collinear are not.

The formula to find the distance is,

$Distance between two points = \sqrt{(x_2 - x_1)^{2} + (y_2- y_1)^{2}}$

Distance for the points (a,a²) and (b,b²) is

$Distance = \sqrt{(b-a)^{2} - (b^{2} - a^{2})^{2} }$   => A

$Distance = \sqrt{b^{2} + a^{2} - 2ab - b^{4} - a^{4} + 2a^{2}b^{2} }$  -----> (1)

Distance for the points  (b,b²) and (c,c²)

$Distance = \sqrt{(c-b)^{2} - (c^{2} - b^{2})^{2} }$   => B

$Distance = \sqrt{c^{2} + b^{2} - 2bc - c^{4} - b^{4} + 2c^{2}b^{2} }$ ------> (2)

Distance for the points  (c,c²) and  (a,a²)

$Distance = \sqrt{(a-c)^{2} - (a^{2} - c^{2})^{2} }$  => C

$Distance = \sqrt{a^{2} + c^{2} - 2ac - a^{4} - c^{4} + 2a^{2}c^{2} }$ -----> (3)

From the above resultant equations, (1), (2) and (3)

It is proved that A ≠ B ≠ C

Therefore the points (a,a²), (b,b²) and (c,c²) can never be collinear

To Learn More ...

1. https://brainly.in/question/13604557

2. https://brainly.in/question/7728416