Solve the problem.
-Find an equation for the level curve of the function $f ( x , y ) = \sum _ { n = 0 } ^ { \infty } \left( \frac { x } { y } \right) ^ { n }$ that passes through the point $( 1,5 )$.
A) $\left| \frac { x } { y } \right| = \frac { 1 } { 5 }$
B) $x = 5 y$
C) $\left| \frac { x } { y } \right| = 5$
D) $\left| \frac { x } { y } \right| \leq \frac { 1 } { 5 }$

Question

Quizplus · Accepted Answer

The given function is a geometric series with the common ratio $r = \frac{x}{y}$. For the series to converge, $|r| < 1$, and its sum is given by $\frac{1}{1 - r} = \frac{1}{1 - \frac{x}{y}}$. The level curve passing through $(1,5)$ implies $r = \frac{1}{5}$, so the equation for the level curve is $\left| \frac{x}{y} \right| = \frac{1}{5}$.

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