Given that the graph of $y = kx$ passes through the point $(a, b)$ and $\frac{a}{b} = 2$, we need to determine the value of $k$.

1. Since the point $(a, b)$ lies on the line $y = kx$, we can substitute $x = a$ and $y = b$ into the equation: $b = ka$

2. We are also given that $\frac{a}{b} = 2$. This implies: $a = 2b$

3. Substitute $a = 2b$ into the equation $b = ka$: $b = k(2b)$

4. Simplify this equation: $b = 2kb$

5. To solve for $k$, divide both sides by $b$ (assuming $b \neq 0$): $1 = 2k$

6. Finally, solve for $k$: $k = \frac{1}{2}$

Answer:

The value of $k$ is $\frac{1}{2}$.

Would you like more details, or do you have any questions?

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Here are some related questions:

1. What happens to the line's slope if $\frac{a}{b} = \frac{1}{2}$ instead?
2. How does the line $y = kx$ change if $k = -\frac{1}{2}$?
3. What is the significance of $k$ when $y = kx$ passes through the origin?
4. How would the equation change if the point given was in the form $(-a, -b)$?
5. What happens to $k$ if the ratio $\frac{a}{b}$ changes?

Tip: When given points or ratios, substituting them directly into the equation can simplify solving for unknown variables.