A point of the parabola ` y^2 = 18 x ` at which the ordinate increases at twice the rate of abscissa is :

Options

A. `(2,4)`
B. ` (2,-4)`
C. ` (-9/8 ,9/2)`
D. ` (9/8 , (9)/(2))`

Answer

To solve the problem step by step, we will follow the given information and the differentiation of the parabola \( y^2 = 18x \).

### Step 1: Understand the relationship between the rates of change
Given that the ordinate (y) increases at twice the rate of the abscissa (x), we can express this relationship mathematically as:
\[ \frac{dy}{dt} = 2 \frac{dx}{dt} \]

### Step 2: Differentiate the equation of the parabola
The equation of the parabola is given as:
\[ y^2 = 18x \]
We will differentiate both sides with respect to time \( t \):
\[ \frac{d}{dt}(y^2) = \frac{d}{dt}(18x) \]
Using the chain rule, we have:
\[ 2y \frac{dy}{dt} = 18 \frac{dx}{dt} \]

### Step 3: Substitute the relationship into the differentiated equation
From Step 1, we know that \( \frac{dy}{dt} = 2 \frac{dx}{dt} \). We can substitute this into the differentiated equation:
\[ 2y (2 \frac{dx}{dt}) = 18 \frac{dx}{dt} \]
This simplifies to:
\[ 4y \frac{dx}{dt} = 18 \frac{dx}{dt} \]

### Step 4: Cancel \( \frac{dx}{dt} \) (assuming it is not zero)
Assuming \( \frac{dx}{dt} \neq 0 \), we can divide both sides by \( \frac{dx}{dt} \):
\[ 4y = 18 \]
Now, solving for \( y \):
\[ y = \frac{18}{4} = \frac{9}{2} \]

### Step 5: Substitute \( y \) back into the original equation to find \( x \)
Now that we have \( y = \frac{9}{2} \), we substitute this back into the original parabola equation:
\[ \left(\frac{9}{2}\right)^2 = 18x \]
Calculating \( \left(\frac{9}{2}\right)^2 \):
\[ \frac{81}{4} = 18x \]
Now, solving for \( x \):
\[ x = \frac{81}{4 \cdot 18} = \frac{81}{72} = \frac{9}{8} \]

### Final Result
Thus, the point on the parabola where the ordinate increases at twice the rate of the abscissa is:
\[ \left( \frac{9}{8}, \frac{9}{2} \right) \]

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