First slide

Cartesian plane

Question

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in G.P. with the same common ratio, then the points (x₁, y₁), (x₂, y₂) and (x₃,y₃)

Moderate

A

lie on a straight line
B

lie on an ellipse
C

lie on a circle
D

are vertices of a triangle

Solution

$L e t \frac{x_{2}}{x_{1}} = \frac{x_{3}}{x_{2}} = r a n d \frac{y_{2}}{y_{1}} = \frac{y_{3}}{y_{2}} = r$
⇒ x₂ = x₁r, x₃ = x₁r², y₂ = y₁r and y₃ = y₁r².
We have,
$∆ = |\begin{matrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 2 \\ x_{3} & y_{3} & 3 \end{matrix}| = |\begin{matrix} x_{1} & y_{1} & 1 \\ x_{1} r & y_{1} r & 1 \\ x_{1} r^{2} & y_{1} r^{2} & 1 \end{matrix}|$
$= |\begin{matrix} x_{1} & y_{1} & 1 \\ 0 & 0 & 1 - r \\ 0 & 0 & 1 - r \end{matrix}|$
(Applying R₃ → R₃ – rR₂and R₂ → R₂ – rR₁) = 0
( $∴$ R₂ and R₃ are identical)
Thus, (x_1, y₁), (x₂, y₂), (x₃, y₃) lie on a straight line

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