If x_1, x_2, x_3 as well as y_1, y_2, y_3 are | vedclass

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(D) Given that $x_1, x_2, x_3$ and $y_1, y_2, y_3$ are in geometric progression $(GP)$ with the same common ratio $r$. Let $x_1 = a, x_2 = ar, x_3 = a

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collinear

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(D) Given that $x_1, x_2, x_3$ and $y_1, y_2, y_3$ are in geometric progression $(GP)$ with the same common ratio $r$. Let $x_1 = a, x_2 = ar, x_3 = ar^2$ and $y_1 = b, y_2 = br, y_3 = br^2$. The points are $A(a, b), B(ar, br), C(ar^2, br^2)$. To check if they are collinear,we calculate the area of the triangle formed by these points: $	ext{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$ $= \frac{1}{2} |a(br - br^2) + ar(br^2 - b) + ar^2(b - br)|$ $= \frac{1}{2} |abr(1 - r) + abr(r^2 - 1) + abr^2(1 - r)|$ $= \frac{1}{2} |abr - abr^2 + abr^3 - abr + abr^2 - abr^3| = 0$. Since the area of the triangle is $0$,the points are collinear.

If $x_1, x_2, x_3$ as well as $y_1, y_2, y_3$ are in geometric progression with the same common ratio,then the points $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ are

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