If x_1, x_2, x_3 and y_1, y_2, y_3 are in G.P | vedclass

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(A) Let the points be $(x_1, y_1) = (a, b)$,$(x_2, y_2) = (ar, br)$,and $(x_3, y_3) = (ar^2, br^2)$ where $r$ is the common ratio.To check for colline

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Collinear

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(A) Let the points be $(x_1, y_1) = (a, b)$,$(x_2, y_2) = (ar, br)$,and $(x_3, y_3) = (ar^2, br^2)$ where $r$ is the common ratio.To check for collinearity,we calculate the area of the triangle formed by these points using the determinant method:$\Delta = \frac{1}{2} \left| \begin{array}{ccc} a & b & 1 \ ar & br & 1 \ ar^2 & br^2 & 1 \end{array} ight|$Factoring out $r$ from the second row and $r^2$ from the third row:$\Delta = \frac{1}{2} \cdot r \cdot r^2 \left| \begin{array}{ccc} a & b & 1 \ a & b & 1 \ a & b & 1 \end{array} ight| = 0$Since the area is $0$,the points are collinear (lie on a straight line).

If $x_1, x_2, x_3$ and $y_1, y_2, y_3$ are in $G.P.$ with the same common ratio,then the points $(x_1, y_1), (x_2, y_2),$ and $(x_3, y_3)$ are:

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