JEE MAIN - Mathematics (2003 - No. 46)
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 $$\left( {{x_1},{y_1}} \right),\left( {{x_2},{y_2}} \right)$$ and $$\left( {{x_3},{y_3}} \right)$$ :
are vertices of a triangle
lie on a straight line
lie on an ellipse
lie on a circle
Explanation
Taking co-ordinates as
$$\left( {{x \over r},{y \over r}} \right);\left( {x,y} \right)\,\,\& \,\,\left( {xr,yr} \right)$$
Then slope of line joining
$$\left( {{x \over r},{y \over r}} \right),\left( {x,y} \right) = {{y\left( {1 - {1 \over r}} \right)} \over {x\left( {1 - {1 \over r}} \right)}} = {y \over x}$$
and slope of line joining $$(x,y)$$ and $$(xr, yr)$$
$$ = {{y\left( {r - 1} \right)} \over {x\left( {r - 1} \right)}} = {y \over x}$$
$$\therefore$$ $${m_1} = {m_2}$$
$$ \Rightarrow $$ Points lie on the straight line.
$$\left( {{x \over r},{y \over r}} \right);\left( {x,y} \right)\,\,\& \,\,\left( {xr,yr} \right)$$
Then slope of line joining
$$\left( {{x \over r},{y \over r}} \right),\left( {x,y} \right) = {{y\left( {1 - {1 \over r}} \right)} \over {x\left( {1 - {1 \over r}} \right)}} = {y \over x}$$
and slope of line joining $$(x,y)$$ and $$(xr, yr)$$
$$ = {{y\left( {r - 1} \right)} \over {x\left( {r - 1} \right)}} = {y \over x}$$
$$\therefore$$ $${m_1} = {m_2}$$
$$ \Rightarrow $$ Points lie on the straight line.
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