Find the point on the parabola `y^(2)=18x` at which ordinate is 3 times its abscissa.

Answer

To find the point on the parabola \( y^2 = 18x \) where the ordinate (y-coordinate) is three times the abscissa (x-coordinate), we can follow these steps:

### Step 1: Set up the relationship between x and y
Given that the ordinate is three times the abscissa, we can express this relationship mathematically as:
\[ y = 3x \]

### Step 2: Substitute y in the parabola equation
Now, substitute \( y = 3x \) into the parabola equation \( y^2 = 18x \):
\[ (3x)^2 = 18x \]

### Step 3: Simplify the equation
This simplifies to:
\[ 9x^2 = 18x \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 9x^2 - 18x = 0 \]

### Step 5: Factor the equation
We can factor out the common term:
\[ 9x(x - 2) = 0 \]

### Step 6: Solve for x
Setting each factor to zero gives us:
\[ 9x = 0 \quad \text{or} \quad x - 2 = 0 \]
Thus, we find:
\[ x = 0 \quad \text{or} \quad x = 2 \]

### Step 7: Find corresponding y values
Now, we will find the corresponding y values for each x:
1. For \( x = 0 \):
\[ y = 3(0) = 0 \quad \Rightarrow \quad (0, 0) \]
2. For \( x = 2 \):
\[ y = 3(2) = 6 \quad \Rightarrow \quad (2, 6) \]

### Step 8: State the points
The points on the parabola where the ordinate is three times the abscissa are:
\[ (0, 0) \quad \text{and} \quad (2, 6) \]

### Final Answer
The points on the parabola \( y^2 = 18x \) at which the ordinate is three times the abscissa are \( (0, 0) \) and \( (2, 6) \).

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