N.S. condition of passing a fixed point for a function

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N.S. condition of passing a fixed point for a function

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Kunihiko_Chikaya

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Kunihiko_Chikaya

#1 h Sep 6, 2009, 11:27 AM • 1 Y

Y by Adventure10

Let $f(t)$ be a function defined in any real numbers $t$ with $f(0)\neq 0.$ Prove that on the $x-y$ plane, the line $l_t : tx+f(t) y=1$ passes through the fixed point which isn't on the $y$ axis in regardless of the value of $t$ if only if $f(t)$ is a linear function in $t$ .

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#2 h Apr 29, 2025, 11:29 AM

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If we write $l_{t}$ is slope-intercept form, then $y = \frac{1-tx}{f(t)}$ Knowing that $l_{t}$ passes through a fixed point $P(a,b)$ with $a \neq 0$ , we obtain:

$b = \frac{1-at}{f(t)} \Rightarrow f(t) = \frac{1-at}{b}$ ;

of which $f(t)$ is linear.

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