Find the position of the point (6,-6) w.r.to the parabola `y^(2)` = 6x.

Answer

To determine the position of the point (6, -6) with respect to the parabola defined by the equation \( y^2 = 6x \), we will follow these steps:

### Step 1: Identify the equation of the parabola
The given equation of the parabola is:
\[ y^2 = 6x \]

### Step 2: Write the expression for \( S_1 \)
To find the position of the point with respect to the parabola, we need to derive the expression \( S_1 \). We can rewrite the equation of the parabola in the form:
\[ S_1 = y^2 - 6x \]

### Step 3: Substitute the coordinates of the point into \( S_1 \)
Now, we substitute the coordinates of the point (6, -6) into the expression for \( S_1 \):
\[ S_1 = (-6)^2 - 6(6) \]

### Step 4: Calculate \( S_1 \)
Now we perform the calculations:
\[ S_1 = 36 - 36 = 0 \]

### Step 5: Analyze the result
We have three conditions to determine the position of the point with respect to the parabola:
- If \( S_1 > 0 \), the point is outside the parabola.
- If \( S_1 < 0 \), the point is inside the parabola.
- If \( S_1 = 0 \), the point is on the parabola.

Since we found that \( S_1 = 0 \), this means that the point (6, -6) is on the parabola.

### Final Conclusion
The position of the point (6, -6) with respect to the parabola \( y^2 = 6x \) is that it lies on the parabola.

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