To sketch the equation $ay(x^2 - \frac{1}{3}y^2) = C$ in the first quadrant, where $a$ and $C$ are constants, we need to follow several steps. Here's the detailed process:

Step 1: Express the equation in a more convenient form

Start by rewriting the equation for clarity: $a y \left( x^2 - \frac{1}{3} y^2 \right) = C$ Now, solve for $y$: $y \left( x^2 - \frac{1}{3} y^2 \right) = \frac{C}{a}$ This is a non-linear equation, so let's attempt to solve it for specific values of $x$ and $y$.

Step 2: Analyze the equation for different values of $x$

When $x = 0$:

Substitute $x = 0$ into the equation: $a y \left( 0^2 - \frac{1}{3} y^2 \right) = C$ This simplifies to: $- \frac{a}{3} y^3 = C$ Solve for $y$: $y^3 = -\frac{3C}{a}$ Thus, we have $y = \left( -\frac{3C}{a} \right)^{1/3}$.

When $y = 0$:

Substitute $y = 0$ into the equation: $a \cdot 0 \cdot \left( x^2 - \frac{1}{3} \cdot 0^2 \right) = C$ This simplifies to $0 = C$, which implies $C = 0$.

Step 3: Explore the general form for $y > 0$ and $x > 0$

Now let's express $y$ in terms of $x$. From the original equation: $a y \left( x^2 - \frac{1}{3} y^2 \right) = C$ Solve for $y$: $y = \frac{C}{a \left( x^2 - \frac{1}{3} y^2 \right)}$ This equation is implicit, and finding an explicit solution for $y$ in terms of $x$ is not trivial without further assumptions. However, you can still sketch the curve numerically or graphically by considering specific values for $C$ and $a$.

Step 4: General shape and behavior

• In the first quadrant, both $x > 0$ and $y > 0$, and the equation describes a relationship between $x$ and $y$ that is implicitly non-linear.
• As $x$ increases, the term $x^2$ dominates, and the behavior of the curve will tend to behave like a hyperbola or some similar non-linear shape.

Step 5: Sketching the curve

You can start sketching the curve by:

1. Plotting the asymptotes (behavior as $x$ or $y$ approach large values).
2. Using specific values for $x$ and $y$ to plot points.
3. Connecting these points to form a curve.

In the first quadrant, the curve would start near the origin (but not pass through it if $C \neq 0$), then curve upwards and to the right, potentially approaching asymptotic behavior based on the form of the equation.

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Additional Insight:

Would you like a more detailed solution using specific values for $a$ and $C$, or would you prefer to explore the asymptotic behavior or further graphing techniques?

5 Related Questions:

1. How does changing the value of $C$ affect the shape of the curve?
2. What happens to the graph as $a$ increases?
3. Can you find the equation's asymptotes?
4. What is the behavior of the curve when $y$ approaches large values?
5. How does this equation compare to a general hyperbola?

Tip: A good way to understand implicit relationships between variables is to solve for one variable (e.g., $y$) numerically for specific values of the other (e.g., $x$), especially if the algebraic manipulation is complex.