Key Concepts and Formulas

• Related Rates: The rates of change of related quantities are related through differentiation, often with respect to time. If $y = f(x)$ and both $x$ and $y$ are functions of $t$, then $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$.
• Implicit Differentiation: Differentiating an equation where variables are not explicitly isolated. For example, $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$.
• Chain Rule: A fundamental rule in calculus that states $\frac{dy}{dx} = \frac{dy}{dt} / \frac{dx}{dt}$.

Step-by-Step Solution

Step 1: Express the given rate condition mathematically.

The problem states that the ordinate (y-coordinate) increases at twice the rate of the abscissa (x-coordinate). This means $\frac{dy}{dt} = 2 \frac{dx}{dt}$. Our goal is to find the point $(x, y)$ on the parabola where this condition holds.

Why this step? This step translates the problem's verbal condition into a mathematical equation, which is the foundation for solving the problem.

$\frac{dy}{dt} = 2 \frac{dx}{dt}$

Step 2: Relate the rate condition to $\frac{dy}{dx}$.

We know that $\frac{dy}{dx}$ represents the instantaneous rate of change of $y$ with respect to $x$. From the chain rule, we have $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$. Substituting the given condition $\frac{dy}{dt} = 2 \frac{dx}{dt}$ into this equation, we get:

Why this step? We want to relate the rates of change with respect to time to the derivative $\frac{dy}{dx}$, which can be found from the parabola's equation.

$\frac{dy}{dx} = \frac{2 \frac{dx}{dt}}{\frac{dx}{dt}}$

Assuming $\frac{dx}{dt} \neq 0$ (otherwise, the abscissa isn't changing), we can simplify to:

$\frac{dy}{dx} = 2$

This tells us that the slope of the tangent to the parabola at the desired point is 2.

Step 3: Find $\frac{dy}{dx}$ using implicit differentiation.

We are given the equation of the parabola: $y^2 = 18x$. We need to find an expression for $\frac{dy}{dx}$ in terms of $x$ and $y$. We will differentiate both sides of the equation with respect to $x$.

Why this step? Implicit differentiation is used to find the derivative of $y$ with respect to $x$ when $y$ is not explicitly defined as a function of $x$.

$\frac{d}{dx}(y^2) = \frac{d}{dx}(18x)$

Applying the chain rule on the left side and the power rule on the right side:

$2y \frac{dy}{dx} = 18$

Now, solve for $\frac{dy}{dx}$:

$\frac{dy}{dx} = \frac{18}{2y} = \frac{9}{y}$

Step 4: Find the y-coordinate of the point.

From Step 2, we have $\frac{dy}{dx} = 2$. From Step 3, we have $\frac{dy}{dx} = \frac{9}{y}$. Setting these equal to each other:

Why this step? Equating the two expressions for the slope allows us to solve for the y-coordinate.

$2 = \frac{9}{y}$

Solving for $y$:

$2y = 9$

$y = \frac{9}{2}$

Step 5: Find the x-coordinate of the point.

Since the point $(x, y)$ lies on the parabola $y^2 = 18x$, we can substitute $y = \frac{9}{2}$ into this equation to find the corresponding $x$-coordinate.

Why this step? This step uses the parabola equation to find the $x$-coordinate corresponding to the $y$-coordinate we just found.

$\left( \frac{9}{2} \right)^2 = 18x$

$\frac{81}{4} = 18x$

Solving for $x$:

$x = \frac{81}{4 \times 18} = \frac{81}{72} = \frac{9 \times 9}{9 \times 8} = \frac{9}{8}$

Step 6: State the coordinates of the required point.

The coordinates of the point are $\left( \frac{9}{8}, \frac{9}{2} \right)$.

Comparing this with the given options, we find that it matches option (A).

Common Mistakes & Tips

• Forgetting the chain rule: When implicitly differentiating, remember to apply the chain rule to terms involving $y$. For example, $\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$.
• Incorrectly interpreting the rate condition: Ensure you correctly translate the given relationship between the rates of change into a mathematical equation.
• Assuming $\frac{dx}{dt} = 0$: The problem implies that both $x$ and $y$ are changing, so $\frac{dx}{dt}$ cannot be zero.

Summary

We started by translating the given rate condition into a mathematical equation, $\frac{dy}{dt} = 2 \frac{dx}{dt}$. Using the chain rule, we found that $\frac{dy}{dx} = 2$. We then used implicit differentiation on the parabola's equation to find another expression for $\frac{dy}{dx}$, which was $\frac{9}{y}$. Equating these two expressions allowed us to solve for $y$, and substituting this value back into the parabola's equation gave us the corresponding $x$ value. This led us to the point $\left( \frac{9}{8}, \frac{9}{2} \right)$.

Final Answer

The final answer is $\boxed{\left( \frac{9}{8}, \frac{9}{2} \right)}$, which corresponds to option (A).