JEE MAIN - Mathematics (2005 - No. 55)
If non zero numbers $$a, b, c$$ are in $$H.P.,$$ then the straight line $${x \over a} + {y \over b} + {1 \over c} = 0$$ always passes through a fixed point. That point is :
$$(-1,2)$$
$$(-1, -2)$$
$$(1, -2)$$
$$\left( {1, - {1 \over 2}} \right)$$
Explanation
$$a,b,c$$ are in $$H.P. \Rightarrow {1 \over a}.{1 \over b},{1 \over c}$$ are in $$A.P.$$
$$ \Rightarrow {2 \over b} = {1 \over a} + {1 \over c}$$
$$ \Rightarrow {1 \over a} - {2 \over b} + {1 \over c} = 0$$
$$\therefore$$ $${x \over a} + {y \over a} + {1 \over c} = 0$$ passes through $$\left( {1, - 2} \right)$$
$$ \Rightarrow {2 \over b} = {1 \over a} + {1 \over c}$$
$$ \Rightarrow {1 \over a} - {2 \over b} + {1 \over c} = 0$$
$$\therefore$$ $${x \over a} + {y \over a} + {1 \over c} = 0$$ passes through $$\left( {1, - 2} \right)$$
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