The point on the parabola `y ^(2) = 36x` whose ordinate is three times the abscissa, is

Options

A. `(4, 12)`
B. `(6,2)`
C. `(2,6)`
D. `(1,3)`

Answer

To find the point on the parabola \( y^2 = 36x \) where the ordinate (y-coordinate) is three times the abscissa (x-coordinate), we can follow these steps:

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

### Step 2: Substitute into the parabola equation
Now, we substitute \( y = 3x \) into the equation of the parabola \( y^2 = 36x \):
\[ (3x)^2 = 36x \]

### Step 3: Simplify the equation
This simplifies to:
\[ 9x^2 = 36x \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 9x^2 - 36x = 0 \]

### Step 5: Factor the equation
We can factor out \( 9x \):
\[ 9x(x - 4) = 0 \]

### Step 6: Solve for x
Setting each factor to zero gives:
\[ 9x = 0 \quad \text{or} \quad x - 4 = 0 \]
Thus, we have:
\[ x = 0 \quad \text{or} \quad x = 4 \]

### Step 7: Find corresponding y values
Now we find the corresponding y values for each x:
1. For \( x = 0 \):
\[ y = 3(0) = 0 \quad \Rightarrow \quad (0, 0) \]
2. For \( x = 4 \):
\[ y = 3(4) = 12 \quad \Rightarrow \quad (4, 12) \]

### Step 8: Final points
Thus, the points on the parabola are:
\[ (0, 0) \quad \text{and} \quad (4, 12) \]

### Conclusion
The points on the parabola \( y^2 = 36x \) whose ordinate is three times the abscissa are \( (0, 0) \) and \( (4, 12) \).

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