If   x_{1},x_{2},x_{3}\: and \: y_{1},y_{2},y_{3}   are both in G.P. with the same common ratio,

then the points \left ( x_{1}, y_{1} \right ),\left ( x_{2}, y_{2} \right ),\left ( x_{3}, y_{3} \right )

  • Option 1)

    lie on an ellipse

  • Option 2)

    lie on a circle

  • Option 3)

    are vertices of a triangle

  • Option 4)

    lie on a straight line.

 

Answers (1)

As we learnt in 

Common ratio of a GP (r) -

The ratio of two consecutive terms of a GP

- wherein

eg: in 2, 4, 8, 16, - - - - - - -

r = 2

and in 100, 10, 1, 1/10 - - - - - - -

r = 1/10

 

 Let r be common ratio

\therefore x_{2}=x_{1}r          y_{2}= y_{1}r

    x_{3}= x_{1}r^{2}         y_{3}= y_{1}r^{2}

Therefore, area of triangle = \begin{vmatrix} x_{1} & y_{1} & 1\\ x_{2} & y_{2} & 1\\ x_{3} & y_{3} & 1 \end{vmatrix} = \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}

= x_{1}y_{1}\begin{vmatrix} 1 & 1 & 1\\ r & r & 1\\ r^{a} & r^{2} & 1 \end{vmatrix}= 0             [Since two columns are equal]

So, they lie on a straight line.


Option 1)

lie on an ellipse

This option is incorrect.

Option 2)

lie on a circle

This option is incorrect.

Option 3)

are vertices of a triangle

This option is incorrect.

Option 4)

lie on a straight line.

This option is correct.

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