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 (1) vertices of an equilateral triangle (2) vertices of a right angled triangle (3) vertices of a right angled isosceles triangle (4) collinear

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in geometric progression with the same common ratio, then the points (

Exercise 7.5 · Q18

x_1, x_2, x_3

(1) vertices of an equilateral triangle

(2) vertices of a right angled triangle

(3) vertices of a right angled isosceles triangle

(4) collinear

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Substitute the GP terms into the area determinant.

This tests collinearity via determinants.

Step 1. Let the common ratio be $r$ : $x_2 = x_1 r,\ x_3 = x_1 r^2$ and $y_2 = y_1 r,\ y_3 = y_1 r^2$ .

Step 2. The area $= \tfrac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_1 r & y_1 r & 1 \\ x_1 r^2 & y_1 r^2 & 1 \end{vmatrix}$ . Compute the cross terms: $(P_2 - P_1) = (x_1(r-1),\ y_1(r-1))$ and $(P_3 - P_1) = (x_1(r^2-1),\ y_1(r^2-1))$ …

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Q.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

Concept understanding — Area of a Triangle and Collinearity using Determinants

Q.If x1,x2,x3x_1, x_2, x_3x1​,x2​,x3​ as well as y1,y2,y3y_1, y_2, y_3y1​,y2​,y3​ are in geometric progression with the same common ratio, then the points (x1,y1), (x2,y2), (x3,y3)(x_1, y_1),\ (x_2, y_2),\ (x_3, y_3)(x1​,y1​), (x2​,y2​), (x3​,y3​) are

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Q.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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