A point on the parabola `y^2=18 x` at which the ordinate increases at twice the rate of the abscissa is (a) (2,6) (b) `(2,-6)` `(9/8,-9/2)` (d) `(9/8,9/2)`

Options

A. `(2,6)`
B. `(2,-6)`
C. `((9)/(8),(9)/(2))`
D. `((9)/(8),(9)/(2))`

Answer

To solve the problem, we need to find a point on the parabola defined by the equation \( y^2 = 18x \) where the rate of change of the ordinate (y-coordinate) is twice that of the abscissa (x-coordinate).

### Step-by-Step Solution:

1. **Understand the given information**:
We are given the parabola \( y^2 = 18x \) and the condition that the ordinate increases at twice the rate of the abscissa. This can be mathematically expressed as:
\[ \frac{dy}{dt} = 2 \frac{dx}{dt} \]

2. **Differentiate the parabola**:
To find the relationship between \( x \) and \( y \), we differentiate the equation \( y^2 = 18x \) with respect to \( t \):
\[ \frac{d}{dt}(y^2) = \frac{d}{dt}(18x) \]
Using the chain rule, we get:
\[ 2y \frac{dy}{dt} = 18 \frac{dx}{dt} \]

3. **Substitute the relationship**:
From the condition we derived, substitute \( \frac{dy}{dt} = 2 \frac{dx}{dt} \) into the differentiated equation:
\[ 2y(2 \frac{dx}{dt}) = 18 \frac{dx}{dt} \]
Simplifying this gives:
\[ 4y \frac{dx}{dt} = 18 \frac{dx}{dt} \]

4. **Assuming \( \frac{dx}{dt} \neq 0 \)**:
We can divide both sides by \( \frac{dx}{dt} \) (since it is not zero):
\[ 4y = 18 \]
Thus, we find:
\[ y = \frac{18}{4} = \frac{9}{2} \]

5. **Find the corresponding \( x \)**:
Now that we have \( y = \frac{9}{2} \), we can find \( x \) by substituting \( y \) back into the original parabola equation:
\[ \left(\frac{9}{2}\right)^2 = 18x \]
Calculating \( \left(\frac{9}{2}\right)^2 \):
\[ \frac{81}{4} = 18x \]
To solve for \( x \), we multiply both sides by \( \frac{1}{18} \):
\[ x = \frac{81}{4} \cdot \frac{1}{18} = \frac{81}{72} = \frac{9}{8} \]

6. **Conclusion**:
The point on the parabola is:
\[ \left(\frac{9}{8}, \frac{9}{2}\right) \]

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