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
B
lie on a circle
C
are vertices of triangle
D
lie on a straight line

Solution

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