If non-zero numbers a,b,c are in H.P., then the straight line xa+ yb+1c=0 always passes th…

Question

If non-zero numbers $a,b,c$ are in H.P., then the straight line $\dfrac xa+\dfrac yb+\dfrac1c=0$ always passes through a fixed point. That point is

Accepted Answer

$a,b,c$ in H.P. $\Rightarrow\dfrac1a,\dfrac1b,\dfrac1c$ in A.P. $\Rightarrow\dfrac2b=\dfrac1a+\dfrac1c\Rightarrow\dfrac1a-\dfrac2b+\dfrac1c=0$.
Comparing with $\dfrac xa+\dfrac yb+\dfrac1c=0$ evaluated at $(x,y)=(1,-2)$: $\dfrac1a-\dfrac2b+\dfrac1c=0$ --- always true, so the line passes through $(1,-2)$.

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