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Question

If $$x_{1},x_{2},x_{3}$$ and $$y_{1},y_{2},y_{3}$$ are in GP with same common ratio, then $$\left ( x_{1},y_{1} \right ),\left ( x_{2},y_{2}\right ),\left ( x_{3},y_{3} \right )$$


A
lie on an ellipse
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B
lie on a circle
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C
are vertices of triangle
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D
lie on a straight line
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Solution

The correct option is C lie on a straight line
Let $$\displaystyle x_{1}=a \therefore x_{2}=ar,x_{3}=ar^{2} $$ and 

$$\displaystyle y_{1}=b$$     $$\therefore y_{2}=br,y_{3}=br^{2} $$ 

Now $$\displaystyle A(a,b),B(ar,br),C(ar^{2},br^{2})$$ 

Now slope of $$\displaystyle AB=\frac{b(1-r)}{a(1-r)}=\frac{b}{a} $$ and 

slope of $$\displaystyle BC =\frac{br(1-r)}{ar(1-r)} =\frac{b}{a}$$ 

as slope of $$\displaystyle AB=$$ slope of $$BC$$ 

$$\displaystyle \therefore AB \parallel BC $$ but point $$B$$ is common so
 
$$A, B, C$$ are collinear.

Mathematics

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